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    "text": "Metadados\nOutra opção para a capa animada.\nEste livro é uma publicação da Editora Moan, Foz do Iguaçu - PR, Brasil. Seu identificador é ark:68745/eMR8J. A versão física (impressa) deste livro possui ark:68745/eMR8J.4N e isbn:9786599140440.\nPara saber o que é e como usar um identificador ARK, acesse: https://n2t.net/ark:68745/eMT4d/posts/ark_o_que_e.\nConfira, abaixo, os metadados completos deste livro, conforme registro ARK. Você pode escolher a versão JSON ou YAML.",
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    "text": "Como Citar\nBibTeX:\n\n@book{guzzo_s_m_2021,\n  author = {Guzzo, Sandro Marcos},\n  title = {As funções trigonométricas circulares e hiperbólicas},\n  year = {2021},\n  publisher = {Editora Moan},\n  address = {Foz do Iguaçu},\n  url = {https://livro.online/ark:68745/eMR8J},\n  urldate = {A data que vc acessou no padrão yyyy-mm-dd},\n}\n\nPara atribuição, cite este trabalho como:\n\nGUZZO, S. M. As funções trigonométricas circulares e hiperbólicas. Foz do Iguaçu: Editora Moan, 2021. Disponível em: https://livro.online/ark:68745/eMR8J. Acesso em: dd mmm(3 primeiras letras). yyyy.",
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    "text": "Versão Impressa\nCompre a versão impressa e ajude a manter este projeto com livros gratuitos online. Mantenha o conhecimento livre!\nAssim que o livro físico estiver disponível para venda, ele aparecerá aqui1.",
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    "text": "Direitos Autorais\n© Sandro Marcos Guzzo e a Editora Moan, 2021\nTodos os direitos reservados",
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    "text": "Se algum link de compra estiver quebrado, por favor, nos avise no e-mail editora@livro.online ou pelo whatsapp +55 (45) 9 3505-0721.↩︎",
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    "text": "Meus agradecimentos às pessoas que ajudaram na elaboração deste texto. Em especial:\n    À minha mulher Sandra Maria Tieppo pelo apoio constante e o primeiro trabalho de revisão e correção.\n    Ao professor Rogério Luis Rizzi pelo material bibliográfico que serviu de base para este texto.\n    Ao professor Ma To Fu pela ideia do capítulo com aplicações e ao professor Emerson Mário Boldo pela revisão desse mesmo capítulo.",
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    "text": "As funções trigonométricas circulares e hiperbólicas fazem parte de qualquer curso de Cálculo Diferencial e Integral. A trigonometria circular, conhecida dos alunos desde o ensino fundamental e médio, é em geral bem fundamentada e organizada. São relações definidas e obtidas no triângulo retângulo ou na circunferência trigonométrica e todas as propriedades e identidades, envolvendo essas funções, são provadas a partir das propriedades da circunferência e dos triângulos.\n\n\nEsse fato já não ocorre com as funções trigonométricas hiperbólicas. Em geral, os livros de Cálculo Diferencial e Integral definem as funções trigonométricas hiperbólicas como soma de funções exponenciais. Para ser mais preciso,\n\n\n\n   e\n\n\n\nTodas as propriedades envolvendo as funções trigonométricas hiperbólicas são então deduzidas a partir dessas igualdades e de propriedades das funções exponenciais e logarítmicas. Também, as fórmulas de derivação são obtidas usando as fórmulas de derivação para a função exponencial.\n\n\nAssim como a trigonometria circular, a trigonometria hiperbólica é também construída e fundamentada. Não sobre a circunferência, mas sobre a hipérbole trigonométrica. As propriedades dessas funções são então consequências de propriedades algébricas e geométricas dessa hipérbole.\n\n\nEste assunto me deixou curioso por muito tempo até que resolvi procurar mais informações a esse respeito. Hoje, com essas informações localizadas e reunidas, faço este texto com o objetivo de compartilhar os conhecimentos adquiridos nesse assunto. Para tornar o estudo completo, neste texto é abordada também a trigonometria circular, que na maioria dos livros já está bem detalhada.\n\n\nA proposta deste texto é complementar um curso de Cálculo Diferencial e Integral com detalhamento no trato das funções trigonométricas. Do modo que este texto está preparado, esperamos que o leitor, em cada etapa, esteja familiarizado com os conceitos abordados, como por exemplo os conceitos de continuidade, derivação e função inversa.\n\n\nO primeiro capítulo tratará das funções trigonométricas circulares. Construiremos primeiramente a trigonometria\ncircular sobre o círculo trigonométrico com a dedução das principais identidades trigonométricas. Feito isto,\ndefiniremos as funções trigonométricas circulares e estudaremos alguns de seus principais aspectos, como domínio, imagem, gráficos, continuidade e derivadas dessas funções. Para finalizar o primeiro capítulo, estudaremos as funções trigonométricas inversas e suas derivadas.\n\n\nNo segundo capítulo, trataremos da trigonometria hiperbólica. Estudaremos a definição das funções trigonométricas hiperbólicas na hipérbole trigonométrica e as principais identidades trigonométricas hiperbólicas. Analisaremos as funções trigonométricas hiperbólicas, seus gráficos, continuidade e a derivada dessas funções. E por fim, estudaremos as funções trigonométricas hiperbólicas inversas e suas derivadas.\n\n\nO capítulo 3 é dedicado a mostrar que as identidades trigonométricas hiperbólicas envolvendo as funções exponenciais são verdadeiras. Obteremos, ainda, identidades logarítmicas para as funções trigonométricas hiperbólicas inversas e com o auxílio da álgebra de números complexos, obteremos identidades similares para as funções trigonométricas circulares.\n\n\nNo capítulo 4, são apresentadas algumas aplicações das funções trigonométricas, tanto as circulares quanto as\nhiperbólicas. Para melhor compreensão deste capítulo, recomendamos ao leitor o conhecimento de alguns conceitos\nfísicos.\n\n\nAo final de cada seção, apresentaremos uma tabela resumida com os principais resultados obtidos naquela seção. Essas tabelas tem o objetivo de facilitar a busca de informações desejadas por parte do leitor.\n\n\n\n\n\n\n\n\n\n\n\n\n27. Abril 2021\n\n\nSandro Marcos Guzzo",
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.1 A trigonometria circular",
    "text": "1.1 A trigonometria circular\n\n            \n            A trigonometria surgiu do estudo das medidas de um triângulo retângulo. A palavra trigonometria significa, em grego, medidas dos lados do triângulo. Dado um triângulo retângulo, marca-se um dos ângulos não reto e as relações trigonométricas atribuídas a esse ângulo são as seis razões possíveis envolvendo as medidas da hipotenusa, do cateto oposto (ao ângulo marcado) e do cateto adjacente (ao ângulo marcado). \n            \n            \n                \n                    \n                    \n                    Ver maior | Referência\n                    \n                    \n                    \n                    \n                \n\n        &lt;p class=\" unidade\" id=\"1P3\" title=\"1P3\"&gt;\n        Observando o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img12.svg\" alt=\"$OPA$\" loading=\"lazy\"&gt;&lt;/span&gt; da figura anterior e sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img14.svg\" alt=\"$P\\hat{O}A$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos as seis relações trigonométricas associadas a este ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;, dadas por\n        &lt;/p&gt;\n\n\n\n        &lt;div class=\"mathdisplay unidade\" id=\"1P4\" title=\"1P4\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img15.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.94ex; vertical-align: -1.98ex; \" src=\"img/img16.svg\" alt=\"$\\displaystyle = \\frac{\\text{cateto oposto}}{\\text{hipotenusa}} = \\frac{PA}{OA},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img17.svg\" alt=\"$\\displaystyle \\cos \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.98ex; \" src=\"img/img18.svg\" alt=\"$\\displaystyle = \\frac{\\text{cateto adjacente}}{\\text{hipotenusa}} = \\frac{OP}{OA},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img19.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.99ex; vertical-align: -2.02ex; \" src=\"img/img20.svg\" alt=\"$\\displaystyle = \\frac{\\text{cateto oposto}}{\\text{cateto adjacente}} = \\frac{PA}{OP},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img21.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.98ex; \" src=\"img/img22.svg\" alt=\"$\\displaystyle = \\frac{\\text{cateto adjacente}}{\\text{cateto oposto}} = \\frac{OP}{PA},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img23.svg\" alt=\"$\\displaystyle \\sec \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.99ex; vertical-align: -2.02ex; \" src=\"img/img24.svg\" alt=\"$\\displaystyle = \\frac{\\text{hipotenusa}}{\\text{cateto adjacente}} = \\frac{OA}{OP},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img25.svg\" alt=\"$\\displaystyle \\csc \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.98ex; \" src=\"img/img26.svg\" alt=\"$\\displaystyle = \\frac{\\text{hipotenusa}}{\\text{cateto oposto}} = \\frac{OA}{PA},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n    \n\n\n        &lt;p class=\" unidade\" id=\"1P5\" style=\"text-indent: 0px !important;\" title=\"1P5\"&gt;\n        e chamadas respectivamente de seno, cosseno, tangente, cotangente, secante e cossecante de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;. É comum ainda representar a cossecante do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; por &lt;!-- MATH\n         ${\\mathrm{cossec}}\\alpha$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img27.svg\" alt=\"${\\mathrm{cossec}}\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; e a tangente de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; por &lt;!-- MATH\n         $\\tan \\alpha$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img28.svg\" alt=\"$\\tan \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;. A notação &lt;!-- MATH\n         $\\tan \\alpha$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img28.svg\" alt=\"$\\tan \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; é a notação padrão para a tangente no sistema de medida estadunidense e, por isso, muitas calculadoras usam essa notação.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P6\" title=\"1P6\"&gt;\n        Podemos pensar que à medida que o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; varia, variam também as razões entre os lados desse triângulo e, consequentemente, variam as relações trigonométricas associadas ao ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;. Dessa forma, podemos pensar que as relações trigonométricas são dadas em função do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P7\" title=\"1P7\"&gt;\n        Observemos que é imediato dessas definições que\n        &lt;/p&gt;\n\n\n\n        &lt;div class=\"mathdisplay unidade\" id=\"1P8\" title=\"1P8\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img19.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.06ex; vertical-align: -1.57ex; \" src=\"img/img29.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}\\alpha}{\\cos \\alpha},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img21.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img30.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm {tg}}\\alpha} = \\frac{\\cos \\alpha}{{\\mathrm {sen}}\\alpha},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img23.svg\" alt=\"$\\displaystyle \\sec \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img31.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {tg}}\\alpha}{{\\mathrm {sen}}\\alpha} = \\frac{1}{\\cos \\alpha},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img25.svg\" alt=\"$\\displaystyle \\csc \\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img32.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {ctg}}\\alpha}{\\cos \\alpha} = \\frac{1}{{\\mathrm {sen}}\\alpha}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n\n\n        &lt;p class=\" unidade\" id=\"1P9\" title=\"1P9\"&gt;\n        Queremos então construir as funções trigonométricas, que a cada ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; associam o seno, o cosseno, a tangente, a cotangente, a secante e a cossecante de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos alguns problemas nesse sentido. Primeiro quando pensamos em função, pensamos em domínio, imagem, gráfico da função entre outras propriedades. Um dos problemas é que o sistema cartesiano, utilizado para representar graficamente uma função, possui uma escala de medida baseada no comprimento. Por outro lado, um ângulo é tradicionalmente medido em graus. Precisamos utilizar um sistema de medida de ângulos compatível com o sistema cartesiano. Além disso, a trigonometria em um triângulo retângulo somente pode levar em conta ângulos de amplitude entre 0 e 90 graus (exluindo-se esses dois) e, no caso de funções, queremos estender ao máximo o domínio de definição, pretendendo, inclusive, calcular o valor das seis razões trigonométricas quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; possui uma medida negativa.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P10\" title=\"1P10\"&gt;\n        Vamos construir então o aparato compatível para o desenvolvimento dos nossos estudos.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P11\" title=\"1P11\"&gt;\n        A trigonometria circular é construída sobre uma circunferência unitária, isto é, de raio 1, centrada na origem, cuja equação é &lt;!-- MATH\n         $x^{2} + y^{2} = 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img33.svg\" alt=\"$x^{2} + y^{2} = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Essa circunferência é chamada de circunferência trigonométrica&lt;a name=\"182\"&gt;&lt;/a&gt;, ou círculo trigonométrico&lt;a name=\"183\"&gt;&lt;/a&gt;.\n        &lt;/p&gt;\n\n            \n                \n                    \n                    \n                        Ver maior | Referência\n                    \n                    \n                    \n                    \n                    \n\n        &lt;p class=\" unidade\" id=\"1P12\" title=\"1P12\"&gt;\n        Nessa circunferência, convencionamos que:\n        \n        &lt;/p&gt;&lt;table width=\"90%\" style=\"margin-top: 30px; margin-bottom: 30px;;\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;-&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;O ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img34.svg\" alt=\"$V=(1,0)$\" loading=\"lazy\"&gt;&lt;/span&gt; é a origem de quaisquer arcos a serem marcados na circunferência.\n        &lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;-&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;A cada ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; da circunferência, correspondem um arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; e um ângulo associado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img37.svg\" alt=\"$V\\hat{O}A$\" loading=\"lazy\"&gt;,&lt;/span&gt; marcado no sentido anti-horário, ao qual é atribuída uma medida positiva.\n        &lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;-&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;A cada ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; da circunferência, correspondem também um arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; e um ângulo associado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img37.svg\" alt=\"$V\\hat{O}A$\" loading=\"lazy\"&gt;,&lt;/span&gt; marcado no sentido horário, ao qual é atribuída uma medida negativa.\n        &lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;\n        &lt;p&gt;&lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P13\" title=\"1P13\"&gt;\n        A medida &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; que atribuiremos a esse ângulo é baseada em algum dos sistemas de medidas de ângulos conhecidos. São três os sistemas de medidas de ângulo mais difundidos: graus, grados e radianos.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P14\" title=\"1P14\"&gt;\n        O sistema grado&lt;a name=\"192\"&gt;&lt;/a&gt; é o menos utilizado e consiste em dividir a circunferência trigonométrica em 400 partes iguais, cada fração chamada de 1 grado. É um sistema baseado em escala decimal. Os eixos coordenados dividem portanto a circunferência em 4 partes cada uma com medida 100 grados.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P15\" title=\"1P15\"&gt;\n        O sistema grau&lt;a name=\"193\"&gt;&lt;/a&gt; é bastante conhecido dos estudantes de ensino médio e fundamental. Consiste em dividir a circunferência trigonométrica em 360 partes iguais, cada uma dessas partes chamada de 1 grau. Os eixos coordenados dividem então a circunferência em 4 partes iguais de medida 90 graus cada uma.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P16\" title=\"1P16\"&gt;\n        Relata-se que o sistema grau surgiu por volta de 4000 a.C. com os egípcios. Eles desejavam construir um calendário e, para isso, criaram um círculo com marcas onde poderiam contar os dias do ano. Quando um ano se passasse, a contagem deveria voltar ao ponto de partida, para o início de uma nova contagem. Naquela época, acreditavam que o Sol é que girava em torno da Terra e acreditavam que esta volta completa durava 360 dias. Então construíram uma circunferência dividida em 360 partes iguais e a cada dia que se passava, o marcador avançava &lt;!-- MATH\n         $\\frac{1}{360}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.86ex; \" src=\"img/img38.svg\" alt=\"$\\frac{1}{360}$\" loading=\"lazy\"&gt;&lt;/span&gt; da circunferência.\n        Algumas destas informações foram atualizadas com o tempo. Hoje sabe-se com mais precisão o tempo que a Terra leva para dar uma volta completa em torno do Sol, mas a divisão da circunferência em 360 partes iguais já havia sido consolidada.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P17\" title=\"1P17\"&gt;\n        O sistema radiano&lt;a name=\"196\"&gt;&lt;/a&gt; é o mais adequado para o nosso estudo. Como sabemos, o comprimento de uma\n        circunferência é calculado por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img39.svg\" alt=\"$2\\pi r$\" loading=\"lazy\"&gt;&lt;/span&gt;, sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; o raio da circunferência. Isto significa que a circunferência trigonométrica possui comprimento igual a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. Os eixos coordenados dividem portanto essa circunferência em 4 arcos, de comprimento &lt;!-- MATH\n         $\\frac{\\pi}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; cada um. A cada ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; da circunferência, a medida do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; será exatamente a medida do seu comprimento, considerada negativa, se o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; estiver marcado no sentido horário.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P18\" title=\"1P18\"&gt;\n        Dessa forma, a cada ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre a circunferência está associado um arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; e para este arco, um ângulo de medida positiva e um ângulo de medida negativa. Queremos agora a partir de uma medida, positiva ou negativa, determinar um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; cujo arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; e cujo ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img37.svg\" alt=\"$V\\hat{O}A$\" loading=\"lazy\"&gt;&lt;/span&gt; estejam associados a esta medida.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P19\" title=\"1P19\"&gt;\n        A cada número real &lt;!-- MATH\n         $0 \\leq u &lt; 2\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img43.svg\" alt=\"$0 \\leq u &lt; 2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; está associado um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; da circunferência de forma que o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt;, marcado no sentido anti-horário, mede &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; unidades de comprimento. A partir do número real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, digamos &lt;!-- MATH\n         $u \\in [2\\pi, 4\\pi)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img44.svg\" alt=\"$u \\in [2\\pi, 4\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;, ainda podemos marcar o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; na circunferência, no sentido anti-horário, porém, o comprimento do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; é &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img45.svg\" alt=\"$(u-2\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P20\" title=\"1P20\"&gt;\n        Convencionamos então que a cada número real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img46.svg\" alt=\"$u \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; está associado um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; da circunferência, de forma que o comprimento do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt;, marcado no sentido anti-horário, mede &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img47.svg\" alt=\"$(u-2k\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; para algum &lt;!-- MATH\n         $k \\in \\mathbb{Z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img49.svg\" alt=\"$k \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mais precisamente, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\" loading=\"lazy\"&gt;&lt;/span&gt; satisfaz &lt;!-- MATH\n         $2k\\pi \\leq u &lt; 2(k+1)\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img51.svg\" alt=\"$2k\\pi \\leq u &lt; 2(k+1)\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. Em outras palavras, a medida do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; descontando-se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\" loading=\"lazy\"&gt;&lt;/span&gt; voltas completas na circunferência.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P21\" title=\"1P21\"&gt;\n        Admitindo convenções similares para a marcação de arcos no sentido horário, temos que a cada número real negativo, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img52.svg\" alt=\"$u &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt; corresponde um único ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; na circunferência de forma que o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img54.svg\" alt=\"$VB$\" loading=\"lazy\"&gt;&lt;/span&gt; mede &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img47.svg\" alt=\"$(u-2k\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; para algum &lt;!-- MATH\n         $k \\in \\mathbb{Z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img55.svg\" alt=\"$k&lt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Em outras palavras, a medida do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img54.svg\" alt=\"$VB$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; adicionando-se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img56.svg\" alt=\"$(-k)$\" loading=\"lazy\"&gt;&lt;/span&gt; voltas completas na circunferência.\n        \n         &lt;/p&gt;\n        \n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"202\"&gt;&lt;/a&gt;\n        &lt;table id=\"1I3\" title=\"1I3\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.3:&lt;/strong&gt;\n        Ângulos positivo e negativo.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/./angposneg.png\" alt=\"Image angposneg\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P22\" title=\"1P22\"&gt;\n        Dessa forma, temos que a cada &lt;!-- MATH\n         $u \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, também chamado ângulo radiano &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, corresponde um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; na circunferência de modo que o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; mede &lt;!-- MATH\n         $(|u|+2k\\pi)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img58.svg\" alt=\"$(\\vert u\\vert+2k\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; para &lt;!-- MATH\n         $k \\in \\mathbb{Z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt; que satisfaz &lt;!-- MATH\n         $2k\\pi \\leq u &lt; 2(k+1)\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img51.svg\" alt=\"$2k\\pi \\leq u &lt; 2(k+1)\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desse ponto em diante, escreveremos simplesmente que o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou que o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt;, está associado ao ângulo radiano &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou ainda que o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; ou o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img37.svg\" alt=\"$V\\hat{O}A$\" loading=\"lazy\"&gt;&lt;/span&gt; mede &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; radianos.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P23\" title=\"1P23\"&gt;\n        Note que, se &lt;!-- MATH\n         $u \\in [0,2\\pi]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img59.svg\" alt=\"$u \\in [0,2\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt; então a área do setor circular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img60.svg\" alt=\"$OVA$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual a &lt;!-- MATH\n         $\\frac{u}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img61.svg\" alt=\"$\\frac{u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; unidades de área. Dado um ângulo radiano &lt;!-- MATH\n         $u \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, seja &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; o ponto sobre a circunferência tal que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt; tem medida &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. O ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; possui, no sistema cartesiano prefixado, duas coordenadas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img63.svg\" alt=\"$A=(a,b)$\" loading=\"lazy\"&gt;&lt;/span&gt;. O seno de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;a name=\"207\"&gt;&lt;/a&gt; é definido como sendo a ordenada do\n        ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n         &lt;/p&gt;\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm {sen}}u = \\text{ordenada de }A = b,\n        \\end{displaymath}\n         --&gt;\n        \n\n        &lt;div class=\"mathdisplay unidade\" id=\"1P24\" title=\"1P24\"&gt;\n        &lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img64.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}u =$\" loading=\"lazy\"&gt;&nbsp; &nbsp;ordenada de&nbsp;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img65.svg\" alt=\"$\\displaystyle A = b, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        e o cosseno&lt;a name=\"209\"&gt;&lt;/a&gt; de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é definido como sendo a absissa do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P25\" title=\"1P25\"&gt;\n        &lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img66.svg\" alt=\"$\\displaystyle \\cos u =$\" loading=\"lazy\"&gt;&nbsp; &nbsp;absissa de&nbsp;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img67.svg\" alt=\"$\\displaystyle A = a. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P26\" title=\"1P26\"&gt;\n        Outra forma de ver isto, é traçarmos pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; a perpendicular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img68.svg\" alt=\"$AP$\" loading=\"lazy\"&gt;&lt;/span&gt; ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e a perpendicular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img70.svg\" alt=\"$AQ$\" loading=\"lazy\"&gt;&lt;/span&gt; ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;. O seno do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é então o comprimento do segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img72.svg\" alt=\"$OQ$\" loading=\"lazy\"&gt;&lt;/span&gt; (ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img73.svg\" alt=\"$PA$\" loading=\"lazy\"&gt;&lt;/span&gt;) com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se o segmento tiver sentido contrário ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;, entenderemos seu comprimento como negativo. O cosseno do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual ao comprimento do segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img74.svg\" alt=\"$OP$\" loading=\"lazy\"&gt;&lt;/span&gt; (ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img75.svg\" alt=\"$QA$\" loading=\"lazy\"&gt;&lt;/span&gt;) com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img74.svg\" alt=\"$OP$\" loading=\"lazy\"&gt;&lt;/span&gt; estiver orientado contrariamente ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; entenderemos o comprimento como sendo negativo.\n        &lt;/p&gt;\n\n            \n                \n                    \n                    \n                    Ver maior | Referência\n                    \n                    \n                    \n                    \n                    \n\n        &lt;p class=\" unidade\" id=\"1P27\" title=\"1P27\"&gt;\n        Outras quatro razões trigonométricas chamadas respectivamente de tangente, cotangente, secante e cossecante, são definidas por &lt;a name=\"215\"&gt;&lt;/a&gt; &lt;a name=\"216\"&gt;&lt;/a&gt; &lt;a name=\"217\"&gt;&lt;/a&gt; &lt;a name=\"218\"&gt;&lt;/a&gt;\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}, \\qquad {\\mathrm {ctg}}u = \\frac{\\cos u}{{\\mathrm {sen}}u}, \\qquad\n           \\sec u = \\frac{1}{\\cos u} \\qquad \\text{e} \\qquad \\csc u = \\frac{1}{{\\mathrm {sen}}u}. \n        \\end{displaymath}\n         --&gt;\n        \n\n\n        &lt;div class=\"mathdisplay unidade\" id=\"1P28\" title=\"1P28\"&gt;\n        &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img76.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}, \\qquad {\\mathrm {ctg}}u = \\frac{\\cos u}{{\\mathrm {sen}}u}, \\qquad\n        \\sec u = \\frac{1}{\\cos u}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img77.svg\" alt=\"$\\displaystyle \\qquad \\csc u = \\frac{1}{{\\mathrm {sen}}u}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P29\" title=\"1P29\"&gt;\n        Naturalmente, a observação destas razões no triângulo retângulo, acarreta que as razões sejam sempre números positivos. A definição sobre a circunferência trigonométrica estende estes conceitos. Mas também traz alguns problemas. Por exemplo, a abscissa ou a ordenada do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; podem ser nulas, o que acarreta seno ou cosseno de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; igual a zero. Algumas das razões acima não estarão definidas nesses casos.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P30\" title=\"1P30\"&gt;\n        Dados dois ângulos radianos, de medidas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img78.svg\" alt=\"$-u$\" loading=\"lazy\"&gt;&lt;/span&gt;, consideremos os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre a circunferência,\n        associados aos ângulos radianos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img78.svg\" alt=\"$-u$\" loading=\"lazy\"&gt;&lt;/span&gt; respectivamente. Os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt; estão sobre a circunferência e são simétricos um do outro em relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n\n            \n                \n                    \n                    \n                    Ver maior | Referência\n                    \n                    \n                    \n                    \n                    \n\n        &lt;p class=\" unidade\" id=\"1P31\" title=\"1P31\"&gt;\n        Logo,\n        &lt;/p&gt;\n\n\n\n        &lt;div class=\"mathdisplay unidade\" id=\"1P32\" title=\"1P32\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;abscissa de&nbsp;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img80.svg\" alt=\"$\\displaystyle A$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img81.svg\" alt=\"$\\displaystyle =$\" loading=\"lazy\"&gt;&nbsp; &nbsp;abscissa de&nbsp;&lt;img style=\"height: 1.92ex; vertical-align: -0.10ex; \" src=\"img/img82.svg\" alt=\"$\\displaystyle A',$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;ordenada de&nbsp;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img80.svg\" alt=\"$\\displaystyle A$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img83.svg\" alt=\"$\\displaystyle = -($\" loading=\"lazy\"&gt;ordenada de&nbsp;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img84.svg\" alt=\"$\\displaystyle A'),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P33\" style=\"text-indent: 0px !important;\" title=\"1P33\"&gt;ou ainda,&lt;/p&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P34\" title=\"1P34\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img85.svg\" alt=\"$\\displaystyle \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img81.svg\" alt=\"$\\displaystyle =$\" loading=\"lazy\"&gt;&nbsp; &nbsp;abscissa de&nbsp;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img80.svg\" alt=\"$\\displaystyle A$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img81.svg\" alt=\"$\\displaystyle =$\" loading=\"lazy\"&gt;&nbsp; &nbsp;abscissa de&nbsp;&lt;img style=\"height: 1.92ex; vertical-align: -0.10ex; \" src=\"img/img86.svg\" alt=\"$\\displaystyle A'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img87.svg\" alt=\"$\\displaystyle = \\cos(-u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;\n&lt;/table&gt;\n&lt;/div&gt;\n       &lt;span style=\"text-align: left !important;\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img88.svg\" alt=\"$\\displaystyle e$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n       \n &lt;div class=\"mathdisplay unidade\" id=\"1P34-\" title=\"1P34-\"&gt;\n&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img89.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img81.svg\" alt=\"$\\displaystyle =$\" loading=\"lazy\"&gt;&nbsp; &nbsp;ordenada de&nbsp;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img80.svg\" alt=\"$\\displaystyle A$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img83.svg\" alt=\"$\\displaystyle = -($\" loading=\"lazy\"&gt;ordenada de&nbsp;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img90.svg\" alt=\"$\\displaystyle A')$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img91.svg\" alt=\"$\\displaystyle = -{\\mathrm {sen}}(-u).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"1P35\" title=\"1P35\"&gt;\n        Temos então as igualdades\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P36\" title=\"1P36\"&gt;&lt;table&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right; width:45%;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img92.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(-u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img93.svg\" alt=\"$\\displaystyle = -{\\mathrm {sen}}u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &lt;a id=\"senimpar\"&gt;(&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right; width:45%;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img94.svg\" alt=\"$\\displaystyle \\cos(-u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img95.svg\" alt=\"$\\displaystyle = \\cos u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &lt;a id=\"cospar\"&gt;(&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"1P37\" title=\"1P37\"&gt;\n        Note que as coordenadas do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; são então &lt;!-- MATH\n         $A = (\\cos u, {\\mathrm {sen}}u)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img96.svg\" alt=\"$A = (\\cos u, {\\mathrm {sen}}u)$\" loading=\"lazy\"&gt;&lt;/span&gt; e, como o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; está sobre a circunferência, suas coordenadas devem satisfazer a equação da circunferência &lt;!-- MATH\n         $x^{2} + y^{2} = 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img33.svg\" alt=\"$x^{2} + y^{2} = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos então\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P38\" title=\"1P38\"&gt;&lt;a id=\"idfundcirc\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        \\cos^{2}u + {\\mathrm {sen}}^{2} u = (\\cos u)^{2} + ({\\mathrm {sen}}u)^{2} = 1,\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img97.svg\" alt=\"$\\displaystyle \\cos^{2}u + {\\mathrm {sen}}^{2} u = (\\cos u)^{2} + ({\\mathrm {sen}}u)^{2} = 1,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"1P39\" style=\"text-indent: 0px !important;\" title=\"1P39\"&gt;\n        que é conhecida como a relação fundamental da trigonometria (circular)&lt;a name=\"257\"&gt;&lt;/a&gt;.\n        \n        &lt;/p&gt;&lt;p class=\" unidade\" id=\"1P40\" title=\"1P40\"&gt;\n        Assim como no caso de seno e cosseno, podemos também fazer uma visualização geométrica das outras quatro funções trigonométricas.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P41\" title=\"1P41\"&gt;\n        Consideremos, no círculo trigonométrico, a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; paralela ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; e que passa pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img34.svg\" alt=\"$V=(1,0)$\" loading=\"lazy\"&gt;&lt;/span&gt; e a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; paralela ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e que passa pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img99.svg\" alt=\"$W = (0,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;. São duas retas tangentes à circunferência\n        trigonométrica. Dado um ângulo radiano &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; representado pelo arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt;, prolongamos o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img100.svg\" alt=\"$OA$\" loading=\"lazy\"&gt;&lt;/span&gt; até que ele\n        intercepte as retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; respectivamente nos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img8.svg\" alt=\"$N$\" loading=\"lazy\"&gt;&lt;/span&gt;. A tangente do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é o comprimento do\n        segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img102.svg\" alt=\"$VM$\" loading=\"lazy\"&gt;&lt;/span&gt; com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;. A cotangente de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual ao comprimento do segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img103.svg\" alt=\"$WN$\" loading=\"lazy\"&gt;&lt;/span&gt;, com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n\n\n            \n                \n                    \n                    \n                    Ver maior | Referência\n                    \n                    \n                    \n                    \n                \n\n        &lt;p class=\" unidade\" id=\"1P42\" title=\"1P42\"&gt;\n        Nestes termos, notemos que os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img104.svg\" alt=\"$OAP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img105.svg\" alt=\"$OVM$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes e, portanto,\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        VM = \\frac{VM}{OV} = \\frac{AP}{OP} = \\frac{{\\mathrm {sen}}u}{\\cos u} = {\\mathrm  {tg}}u.\n        \\end{displaymath}\n         --&gt;\n        \n\n\n        &lt;div class=\"mathdisplay unidade\" id=\"1P43\" title=\"1P43\"&gt;\n        &lt;img style=\"height: 4.54ex; vertical-align: -1.57ex; \" src=\"img/img106.svg\" alt=\"$\\displaystyle VM = \\frac{VM}{OV} = \\frac{AP}{OP} = \\frac{{\\mathrm {sen}}u}{\\cos u} = {\\mathrm {tg}}u. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P44\" title=\"1P44\"&gt;\n        Também o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img107.svg\" alt=\"$ONW$\" loading=\"lazy\"&gt;&lt;/span&gt; é semelhante ao triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img104.svg\" alt=\"$OAP$\" loading=\"lazy\"&gt;&lt;/span&gt; e, dessa semelhança, temos que\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        WN = \\frac{WN}{WO} = \\frac{OP}{PA} = \\frac{\\cos u}{{\\mathrm {sen}}u} = {\\mathrm {ctg}}u,\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P45\" title=\"1P45\"&gt;\n        &lt;img style=\"height: 4.54ex; vertical-align: -1.57ex; \" src=\"img/img108.svg\" alt=\"$\\displaystyle WN = \\frac{WN}{WO} = \\frac{OP}{PA} = \\frac{\\cos u}{{\\mathrm {sen}}u} = {\\mathrm {ctg}}u, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P46\" style=\"text-indent: 0px !important;\" title=\"1P46\"&gt;\n        e isso significa que as coordenadas de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img8.svg\" alt=\"$N$\" loading=\"lazy\"&gt;&lt;/span&gt; são &lt;!-- MATH\n         $M = (1, {\\mathrm  {tg}}u)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img109.svg\" alt=\"$M = (1, {\\mathrm {tg}}u)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $N = ({\\mathrm {ctg}}u, 1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img110.svg\" alt=\"$N = ({\\mathrm {ctg}}u, 1)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P47\" title=\"1P47\"&gt;\n        Note que se &lt;!-- MATH\n         $u = \\frac{\\pi}{2} \\pm k\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img111.svg\" alt=\"$u = \\frac{\\pi}{2} \\pm k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para &lt;!-- MATH\n         $k \\in \\mathbb{Z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; coincidirá com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img112.svg\" alt=\"$(0,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img113.svg\" alt=\"$(0,-1)$\" loading=\"lazy\"&gt;&lt;/span&gt; e então o prolongamento do segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img100.svg\" alt=\"$OA$\" loading=\"lazy\"&gt;&lt;/span&gt; não intercepta a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e nesses casos, não está definida a tangente de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isso pode também ser observado na expressão &lt;!-- MATH\n         ${\\mathrm  {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.47ex; vertical-align: -0.85ex; \" src=\"img/img114.svg\" alt=\"${\\mathrm {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, pois nos pontos mencionados, temos um denominador nulo. O mesmo ocorre com &lt;!-- MATH\n         ${\\mathrm {ctg}}u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img115.svg\" alt=\"${\\mathrm {ctg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; nos casos em que &lt;!-- MATH\n         $u = \\pm k\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img116.svg\" alt=\"$u = \\pm k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, pois nesses pontos, &lt;!-- MATH\n         ${\\mathrm {sen}}u = 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img117.svg\" alt=\"${\\mathrm {sen}}u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P48\" title=\"1P48\"&gt;\n        Considerando ainda o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, determinado pelo arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt;, traçamos pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt;, tangente a\n        circunferência trigonométrica que passa pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;. Essa reta corta os eixos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; nos pontos que chamaremos, respectivamente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img119.svg\" alt=\"$X$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img120.svg\" alt=\"$Y$\" loading=\"lazy\"&gt;&lt;/span&gt;. A secante do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; (denotada por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img121.svg\" alt=\"$\\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;) é igual ao comprimento do segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img122.svg\" alt=\"$OX$\" loading=\"lazy\"&gt;&lt;/span&gt;, com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e a cossecante de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; (denotada por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img123.svg\" alt=\"$\\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;) é igual ao comprimento do segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img124.svg\" alt=\"$OY$\" loading=\"lazy\"&gt;&lt;/span&gt;, com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n\n\n\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"280\"&gt;&lt;/a&gt;\n        &lt;table id=\"1I7\" title=\"1I7\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.7:&lt;/strong&gt;\n        Secante e cossecante de um ângulo.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/./angseccsc.png\" alt=\"Image angseccsc\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P49\" title=\"1P49\"&gt;\n        Vemos na figura anterior, que os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img104.svg\" alt=\"$OAP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img125.svg\" alt=\"$OAY$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes e dessa semelhança vem\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        OY = \\frac{OY}{OA} = \\frac{AO}{AP} = \\frac{1}{{\\mathrm {sen}}u} = \\csc u.\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P50\" title=\"1P50\"&gt;\n        &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img126.svg\" alt=\"$\\displaystyle OY = \\frac{OY}{OA} = \\frac{AO}{AP} = \\frac{1}{{\\mathrm {sen}}u} = \\csc u. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P51\" title=\"1P51\"&gt;\n        Também os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img104.svg\" alt=\"$OAP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img127.svg\" alt=\"$OAX$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes e, portanto,\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        OX = \\frac{OX}{OA} = \\frac{OA}{OP} = \\frac{1}{\\cos u} = \\sec u,\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P52\" title=\"1P52\"&gt;\n        &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img128.svg\" alt=\"$\\displaystyle OX = \\frac{OX}{OA} = \\frac{OA}{OP} = \\frac{1}{\\cos u} = \\sec u, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        e isso significa que as coordenadas dos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img119.svg\" alt=\"$X$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img120.svg\" alt=\"$Y$\" loading=\"lazy\"&gt;&lt;/span&gt; são, &lt;!-- MATH\n         $X = (\\sec u, 0)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img129.svg\" alt=\"$X = (\\sec u, 0)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $Y = (0, \\csc u)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img130.svg\" alt=\"$Y = (0, \\csc u)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;p class=\" unidade\" id=\"1P53\" title=\"1P53\"&gt;&lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P54\" title=\"1P54\"&gt;\n        Note ainda que se &lt;!-- MATH\n         $u = \\frac{\\pi}{2} \\pm k\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img111.svg\" alt=\"$u = \\frac{\\pi}{2} \\pm k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para &lt;!-- MATH\n         $k \\in \\mathbb{Z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então como antes, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; coincidirá com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img112.svg\" alt=\"$(0,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; ou\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img113.svg\" alt=\"$(0,-1)$\" loading=\"lazy\"&gt;&lt;/span&gt; e a reta tangente à circunferência que passa por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; será paralela ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e não existirá a secante de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembre-se que nestes pontos &lt;!-- MATH\n         $\\cos u = 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img131.svg\" alt=\"$\\cos u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também se &lt;!-- MATH\n         $u = \\pm k\\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img116.svg\" alt=\"$u = \\pm k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, não existirá &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img123.svg\" alt=\"$\\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt; pelo mesmo motivo. Nesses pontos &lt;!-- MATH\n         ${\\mathrm {sen}}u = 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img117.svg\" alt=\"${\\mathrm {sen}}u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P55\" title=\"1P55\"&gt;\n        O que queremos agora é deduzir as fórmulas de soma de arcos para o seno e o cosseno. Consideremos dois ângulos radianos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt;. O ângulo radiano &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; fica determinado pelo arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt; e o ângulo radiano &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; fica denotado pelo arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img54.svg\" alt=\"$VB$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"298\"&gt;&lt;/a&gt;\n        &lt;table id=\"1I8\" title=\"1I8\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.8:&lt;/strong&gt;\n        Ângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/./anguv.png\" alt=\"Image anguv\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P56\" title=\"1P56\"&gt;\n        Baixamos pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; a perpendicular ao segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt;. Essa perpendicular cruza o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; em um ponto que\n        chamaremos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt;. Vamos rotacionar o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img134.svg\" alt=\"$V\\hat{O}B$\" loading=\"lazy\"&gt;&lt;/span&gt; de forma que o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt; coincida com o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img100.svg\" alt=\"$OA$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        Temos então esquema da figura &lt;a href=\"#figangrep\"&gt;1.9&lt;/a&gt;.\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"figangrep\"&gt;&lt;/a&gt;&lt;a id=\"304\"&gt;&lt;/a&gt;\n        &lt;table id=\"1I9\" title=\"1I9\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.9:&lt;/strong&gt;\n        Ângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; reposicionados.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/./anguv2.png\" alt=\"Image anguv2\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P57\" title=\"1P57\"&gt;\n        Nesses termos, o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img54.svg\" alt=\"$VB$\" loading=\"lazy\"&gt;&lt;/span&gt; está agora ascociado ao ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img135.svg\" alt=\"$(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembremos também que o novo triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img136.svg\" alt=\"$OBC$\" loading=\"lazy\"&gt;&lt;/span&gt; é retângulo em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt; e ainda valem as relações\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm {sen}}v = CB \\qquad \\text{e} \\qquad  \\cos v = OC.\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P58\" title=\"1P58\"&gt;\n        &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img137.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}v = CB$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img138.svg\" alt=\"$\\displaystyle \\qquad \\cos v = OC. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P59\" title=\"1P59\"&gt;\n        Pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; baixamos a perpendicular ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; que cruza esse eixo no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt;. Pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; também baixamos a perpendicular ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; que cruza esse eixo no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img6.svg\" alt=\"$Q$\" loading=\"lazy\"&gt;&lt;/span&gt;. Pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt;, baixamos a perpendicular ao segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img139.svg\" alt=\"$BP$\" loading=\"lazy\"&gt;&lt;/span&gt;, que cruza esse segmento em um ponto que chamaremos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt; e a perpendicular ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; que cruza esse eixo em um ponto que chamaremos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img5.svg\" alt=\"$R$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n         &lt;/p&gt;\n        \n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"310\"&gt;&lt;/a&gt;\n        &lt;table id=\"1I10\" title=\"1I10\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.10:&lt;/strong&gt;\n        Relações nos ângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/./anguv3.png\" alt=\"Image anguv3\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P60\" title=\"1P60\"&gt;\n        Temos então\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm {sen}}(u+v) = PB \\qquad \\text{e} \\qquad  \\cos(u+v) = OP,\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P61\" title=\"1P61\"&gt;\n        &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img140.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(u+v) = PB$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img141.svg\" alt=\"$\\displaystyle \\qquad \\cos(u+v) = OP, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P62\" style=\"text-indent: 0px !important;\" title=\"1P62\"&gt;\n        e também,\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm {sen}}u = QA \\qquad \\text{e} \\qquad  \\cos u = OQ.\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P63\" title=\"1P63\"&gt;\n        &lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img142.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}u = QA$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img143.svg\" alt=\"$\\displaystyle \\qquad \\cos u = OQ. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P64\" title=\"1P64\"&gt;\n        Com base na figura, vemos que os triângulos retângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img144.svg\" alt=\"$OQA$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img145.svg\" alt=\"$ORC$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes e, portanto, valem as igualdades\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{OR}{OC} = \\frac{OQ}{OA} = OQ, \\qquad \\text{e} \\qquad \\frac{RC}{OC} = \\frac{QA}{OA} = QA.\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P65\" title=\"1P65\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img146.svg\" alt=\"$\\displaystyle \\frac{OR}{OC} = \\frac{OQ}{OA} = OQ,$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img147.svg\" alt=\"$\\displaystyle \\qquad \\frac{RC}{OC} = \\frac{QA}{OA} = QA. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P66\" title=\"1P66\"&gt;\n        Notemos agora que o triângulo retângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img148.svg\" alt=\"$CMB$\" loading=\"lazy\"&gt;&lt;/span&gt; é também semelhante ao triângulo retângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img145.svg\" alt=\"$ORC$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para mostrar isso, mostraremos que o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img149.svg\" alt=\"$R\\hat{O}C$\" loading=\"lazy\"&gt;&lt;/span&gt; tem a mesma medida do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img150.svg\" alt=\"$C\\hat{B}M$\" loading=\"lazy\"&gt;&lt;/span&gt;. De fato, o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img151.svg\" alt=\"$CM$\" loading=\"lazy\"&gt;&lt;/span&gt; é paralelo ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e então os ângulos alternos internos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img149.svg\" alt=\"$R\\hat{O}C$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img152.svg\" alt=\"$M\\hat{C}O$\" loading=\"lazy\"&gt;&lt;/span&gt; possuem a mesma medida.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P67\" title=\"1P67\"&gt;\n        Então,\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        M\\hat{C}O + M\\hat{C}B = 90^{\\circ} = M\\hat{C}B + C\\hat{B}M,\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P68\" title=\"1P68\"&gt;\n        &lt;img style=\"height: 2.44ex; vertical-align: -0.27ex; \" src=\"img/img153.svg\" alt=\"$\\displaystyle M\\hat{C}O + M\\hat{C}B = 90^{\\circ} = M\\hat{C}B + C\\hat{B}M, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P69\" style=\"text-indent: 0px !important;\" title=\"1P69\"&gt;\n        e, portanto,\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        R\\hat{O}C = M\\hat{C}O = C\\hat{B}M,\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P70\" title=\"1P70\"&gt;\n        &lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img154.svg\" alt=\"$\\displaystyle R\\hat{O}C = M\\hat{C}O = C\\hat{B}M, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"1P71\" style=\"text-indent: 0px !important;\" title=\"1P71\"&gt;\n        donde os triângulos retângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img148.svg\" alt=\"$CMB$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img145.svg\" alt=\"$ORC$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes. Levando em conta a primeira semelhança (entre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img145.svg\" alt=\"$ORC$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img144.svg\" alt=\"$OQA$\" loading=\"lazy\"&gt;&lt;/span&gt;), são semelhantes os triângulos retângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img148.svg\" alt=\"$CMB$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img144.svg\" alt=\"$OQA$\" loading=\"lazy\"&gt;&lt;/span&gt;. Dessa última semelhança e, sabendo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img155.svg\" alt=\"$OA = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{MC}{CB} = \\frac{QA}{OA} = QA \\qquad \\text{e} \\qquad \\frac{MB}{CB} = \\frac{BM}{BC} = \\frac{OQ}{OA} = OQ.\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P72\" title=\"1P72\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img156.svg\" alt=\"$\\displaystyle \\frac{MC}{CB} = \\frac{QA}{OA} = QA$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img157.svg\" alt=\"$\\displaystyle \\qquad \\frac{MB}{CB} = \\frac{BM}{BC} = \\frac{OQ}{OA} = OQ. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P73\" title=\"1P73\"&gt;\n        Segue disto que\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P74\" title=\"1P74\"&gt;&lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right; width: 45%;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img158.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img159.svg\" alt=\"$\\displaystyle = PB = PM + MB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img160.svg\" alt=\"$\\displaystyle = RC + MB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img161.svg\" alt=\"$\\displaystyle = \\frac{RC}{OC} \\cdot OC + \\frac{MB}{CB} \\cdot CB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img162.svg\" alt=\"$\\displaystyle = QA \\cdot OC + OQ \\cdot CB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n     &lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img163.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}u \\cos v + {\\mathrm {sen}}v \\cos u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &lt;a id=\"addarcsen\"&gt;(&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"1P75\" style=\"text-indent: 0px !important;\" title=\"1P75\"&gt;\n        e também,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P76\" title=\"1P76\"&gt;&lt;table&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right; width: 45%;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img164.svg\" alt=\"$\\displaystyle \\cos(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img165.svg\" alt=\"$\\displaystyle = OP = OR - PR$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img166.svg\" alt=\"$\\displaystyle = OR - MC$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img167.svg\" alt=\"$\\displaystyle = \\frac{OR}{OC} \\cdot OC - \\frac{MC}{CB} \\cdot CB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img168.svg\" alt=\"$\\displaystyle = OQ \\cdot OC - QA \\cdot BC$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n     &lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img169.svg\" alt=\"$\\displaystyle = \\cos u \\cos v - {\\mathrm {sen}}u {\\mathrm {sen}}v.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &lt;a id=\"addarccos\"&gt;(&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"1P77\" title=\"1P77\"&gt;\n        As fórmulas, (&lt;a href=\"#addarcsen\"&gt;1.4&lt;/a&gt;) e  (&lt;a href=\"#addarccos\"&gt;1.5&lt;/a&gt;), juntamente com as fórmulas (&lt;a href=\"#senimpar\"&gt;1.1&lt;/a&gt;), (&lt;a href=\"#cospar\"&gt;1.2&lt;/a&gt;) e (&lt;a href=\"#idfundcirc\"&gt;1.3&lt;/a&gt;) são as cinco principais fórmulas da trigonometria circular. Com essas fórmulas, podemos obter outras fórmulas conhecidas, tais como as fórmulas de duplicação de arcos,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P78\" title=\"1P78\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img170.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(2u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img171.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(u+u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img172.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}u \\cos u + {\\mathrm {sen}}u \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n \n     &lt;tr&gt;\n &lt;td&gt;&nbsp;&lt;/td&gt;&lt;td style=\"text-align:left;\"&gt; &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img173.svg\" alt=\"$\\displaystyle = 2 {\\mathrm {sen}}u \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt; &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n     \n    &lt;/tbody&gt;\n&lt;/table&gt;\n&lt;/div&gt;\n \n       &lt;span class=\"MATH\"&gt;e&lt;/span&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"1P78-\" title=\"1P78-\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;\n     \n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img174.svg\" alt=\"$\\displaystyle \\cos(2u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img175.svg\" alt=\"$\\displaystyle = \\cos(u+u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img176.svg\" alt=\"$\\displaystyle = \\cos u \\cos u - {\\mathrm {sen}}u {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img177.svg\" alt=\"$\\displaystyle = \\cos^{2} u - {\\mathrm {sen}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"1P79\" style=\"text-indent: 0px !important;\" title=\"1P79\"&gt;\n        e as fórmulas de diferença de arcos,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P80\" title=\"1P80\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img178.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(u-v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img179.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}u \\cos(-v) + {\\mathrm {sen}}(-v) \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img180.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}u \\cos v - {\\mathrm {sen}}v \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n       \n&lt;/tbody&gt;\n&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;span style=\"text-align: left;\" class=\"MATH\"&gt;e&lt;/span&gt;&lt;/td&gt;\n      \n &lt;div class=\"mathdisplay unidade\" id=\"1P80-\" title=\"1P80-\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img181.svg\" alt=\"$\\displaystyle \\cos(u-v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img182.svg\" alt=\"$\\displaystyle = \\cos u \\cos(-v) - {\\mathrm {sen}}(-v) {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img183.svg\" alt=\"$\\displaystyle = \\cos u \\cos v + {\\mathrm {sen}}v {\\mathrm {sen}}u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"1P81\" title=\"1P81\"&gt;\n        Ainda como exemplo, vamos obter outras identidades trigonométricas. São de fácil demonstração e somente estamos\n        explicitando por motivos de referência futura. Para ser mais preciso, essas identidades serão úteis na seção\n        &lt;a href=\"#secdercircinv\"&gt;1.7&lt;/a&gt;.\n        &lt;/p&gt;\n        \n        &lt;div class=\"unidade\" id=\"1Teo1\" title=\"1Teo1\"&gt;&lt;a id=\"propidsectg\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n        &lt;i&gt;Valem as seguintes identidades trigonométricas circulares\n        &lt;/i&gt;&lt;table width=\"90%\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img184.svg\" alt=\"$(i)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Para todos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que &lt;!-- MATH\n         $u,v,(u+v) \\in \\mathbb{R}-\\{\\frac{\\pi}{2} + k \\pi; \\, k \\in \\mathbb{Z}\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img185.svg\" alt=\"$u,v,(u+v) \\in \\mathbb{R}-\\{\\frac{\\pi}{2} + k \\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P82\" title=\"1P82\"&gt;&lt;a id=\"idtgadd\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        {\\mathrm  {tg}}(u+v) = \\frac{{\\mathrm  {tg}}u + {\\mathrm  {tg}}v}{1- {\\mathrm  {tg}}u {\\mathrm  {tg}}v}.\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.77ex; vertical-align: -2.03ex; \" src=\"img/img186.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}(u+v) = \\frac{{\\mathrm {tg}}u + {\\mathrm {tg}}v}{1- {\\mathrm {tg}}u {\\mathrm {tg}}v}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        &lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img187.svg\" alt=\"$(ii)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Para todo &lt;!-- MATH\n         $u \\in \\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img188.svg\" alt=\"$u \\in \\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P83\" title=\"1P83\"&gt;&lt;a id=\"idtgsec\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        1 + {\\mathrm  {tg}}^{2} u = \\sec^{2} u.\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img189.svg\" alt=\"$\\displaystyle 1 + {\\mathrm {tg}}^{2} u = \\sec^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        &lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img190.svg\" alt=\"$(iii)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Para todo &lt;!-- MATH\n         $u \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img191.svg\" alt=\"$u \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P84\" title=\"1P84\"&gt;&lt;a id=\"idctgcsc\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        {\\mathrm {ctg}}^{2} u + 1 = \\csc^{2} u.\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img192.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}^{2} u + 1 = \\csc^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (&lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n        \n        Para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img184.svg\" alt=\"$(i)$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P85\" title=\"1P85\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img194.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img195.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}(u+v)}{\\cos(u+v)}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.46ex; vertical-align: -1.72ex; \" src=\"img/img196.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}u \\cos v + {\\mathrm {sen}}v \\cos u}{\\cos u \\cos v - {\\mathrm {sen}}u {\\mathrm {sen}}v}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.03ex; vertical-align: -2.30ex; \" src=\"img/img197.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}u \\cos v + {\\mathrm {sen}}v \\cos u}{\\cos u \\cos v (1 - \\frac{{\\mathrm {sen}}u {\\mathrm {sen}}v}{\\cos u \\cos v}) }$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.44ex; vertical-align: -2.30ex; \" src=\"img/img198.svg\" alt=\"$\\displaystyle = \\left( \\frac{{\\mathrm {sen}}u \\cos v + {\\mathrm {sen}}v \\cos u}...\n        ...\\frac{1}{(1 - \\frac{{\\mathrm {sen}}u}{\\cos u} \\frac{{\\mathrm {sen}}v}{\\cos v})}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img199.svg\" alt=\"$\\displaystyle = \\left( \\frac{{\\mathrm {sen}}u}{\\cos u} + \\frac{{\\mathrm {sen}}v}{\\cos v} \\right) \\frac{1}{(1 - {\\mathrm {tg}}u {\\mathrm {tg}}v)}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img200.svg\" alt=\"$\\displaystyle = \\left( {\\mathrm {tg}}u + {\\mathrm {tg}}v \\right) \\frac{1}{(1 - {\\mathrm {tg}}u {\\mathrm {tg}}v)}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.77ex; vertical-align: -2.03ex; \" src=\"img/img201.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {tg}}u + {\\mathrm {tg}}v}{1- {\\mathrm {tg}}u {\\mathrm {tg}}v}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"1P86\" title=\"1P86\"&gt;\n        As demonstrações de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img187.svg\" alt=\"$(ii)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img190.svg\" alt=\"$(iii)$\" loading=\"lazy\"&gt;&lt;/span&gt; são mais rápidas. De fato,\n         &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"1P87\" title=\"1P87\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img202.svg\" alt=\"$\\displaystyle \\sec^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img203.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cos^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.57ex; \" src=\"img/img204.svg\" alt=\"$\\displaystyle = \\frac{\\cos^{2}u + {\\mathrm {sen}}^{2}u}{\\cos^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img205.svg\" alt=\"$\\displaystyle = 1 + {\\mathrm {tg}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n   \n&lt;/tbody&gt;\n&lt;/table&gt;\n&lt;/div&gt;\n\n   &lt;span style=\"text-align: left;\" class=\"MATH\"&gt;e&lt;/span&gt;\n\n   &lt;div class=\"mathdisplay unidade\" id=\"1P87-\" title=\"1P87-\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img206.svg\" alt=\"$\\displaystyle \\csc^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img207.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm {sen}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.57ex; \" src=\"img/img208.svg\" alt=\"$\\displaystyle = \\frac{\\cos^{2}u + {\\mathrm {sen}}^{2}u}{{\\mathrm {sen}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img209.svg\" alt=\"$\\displaystyle = {\\mathrm {ctg}}^{2} u + 1$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n        &lt;table style=\"margin: 0px !important;\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt; e essa demonstração está concluída.&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;",
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.2 Funções trigonométricas circulares",
    "text": "1.2 Funções trigonométricas circulares\n\n            &lt;p class=\" unidade\" id=\"1P88\" title=\"1P88\"&gt;\n            Nesta seção, vamos estudar os aspectos principais das funções trigonométricas circulares. Será conveniente que o leitor possua conhecimentos conceituais sobre domínio, imagem, gráfico e também limites de funções.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P89\" title=\"1P89\"&gt;\n            Admitindo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma variável real, podemos considerar as funções que a cada valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; associam o seno, o\n            cosseno, a tangente, a cotangente, a secante e a cossecante de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, quando existirem. Olhemos uma por uma essas\n            funções.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P90\" title=\"1P90\"&gt;\n            Para cada valor real de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, a função &lt;a name=\"445\"&gt;&lt;/a&gt;\n             &lt;/p&gt;\n\n\n            &lt;div class=\"mathdisplay unidade\" id=\"1P91\" title=\"1P91\"&gt;\n            &lt;!-- MATH\n             \\begin{eqnarray*}\n            f : \\mathbb{R}& \\to & \\mathbb{R}\\\\\n            u & \\mapsto & w = f(u) = {\\mathrm {sen}}u\n            \\end{eqnarray*}\n             --&gt;\n\n            &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n            &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img210.svg\" alt=\"$\\displaystyle f : \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n            &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n            &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img212.svg\" alt=\"$\\displaystyle \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n            &nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n            &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n            &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img215.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n            &nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"1P92\" style=\"text-indent: 0px !important;\" title=\"1P92\"&gt;\n            associa o seno do ângulo (radiano) &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Da seção anterior, temos que esta função assume 0 para os valores &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img216.svg\" alt=\"$u = k \\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também essa função é limitada, assumindo no máximo 1 e no mínimo -1. É uma função periódica de período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. O seu gráfico é como na figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figfsin\"&gt;&lt;/a&gt;&lt;a id=\"450\"&gt;&lt;/a&gt;\n            &lt;table id=\"1I11\" title=\"1I11\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.11:&lt;/strong&gt;\n            Gráfico da função seno circular.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/./fsin.png\" alt=\"Image fsin\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P93\" title=\"1P93\"&gt;\n            Notemos que é uma função contínua (mostraremos isto detalhadamente na próxima seção), não injetora e nem sobrejetora de &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também o comportamento oscilatório para os infinitos, faz com que não existam os limites de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img218.svg\" alt=\"$f(u)$\" loading=\"lazy\"&gt;&lt;/span&gt; quando &lt;!-- MATH\n             $u \\to \\pm \\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.60ex; vertical-align: -0.27ex; \" src=\"img/img219.svg\" alt=\"$u \\to \\pm \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P94\" title=\"1P94\"&gt;\n            A função cosseno &lt;a name=\"453\"&gt;&lt;/a&gt; se comporta de forma similar. Basta notar que &lt;!-- MATH\n             $\\cos(u-\\frac{\\pi}{2}) = {\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img220.svg\" alt=\"$\\cos(u-\\frac{\\pi}{2}) = {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, a função cosseno é apenas um deslocamento horizontal da função seno. Dessa forma, a função &lt;!-- MATH\n             $w = f(u) = \\cos u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img221.svg\" alt=\"$w = f(u) = \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; é também uma função contínua, periódica de período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, que assume máximo 1 e mínimo -1. Os zeros dessa função (os pontos onde a função intercepta o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;) são &lt;!-- MATH\n             $u = \\frac{\\pi}{2} + k\\pi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img222.svg\" alt=\"$u = \\frac{\\pi}{2} + k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. O gráfico fica como na figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figfcos\"&gt;&lt;/a&gt;&lt;a id=\"460\"&gt;&lt;/a&gt;\n            &lt;table id=\"1I12\" title=\"1I12\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.12:&lt;/strong&gt;\n            Gráfico da função cosseno circular.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/./fcos.png\" alt=\"Image fcos\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P95\" title=\"1P95\"&gt;\n            Não é uma função injetora e nem sobrejetora de &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P96\" title=\"1P96\"&gt;\n            Para a função tangente&lt;a name=\"463\"&gt;&lt;/a&gt;, lembremos que &lt;!-- MATH\n             $f(u) = {\\mathrm  {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.52ex; vertical-align: -0.85ex; \" src=\"img/img223.svg\" alt=\"$f(u) = {\\mathrm {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt; e então por se\n            tratar de uma razão, precisamos nos preocupar com os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; que anulam o denominador. Tais pontos não estarão\n            no domínio de definição de &lt;!-- MATH\n             $f(u) = {\\mathrm  {tg}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img224.svg\" alt=\"$f(u) = {\\mathrm {tg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Os valores para os quais &lt;!-- MATH\n             $\\cos u = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img131.svg\" alt=\"$\\cos u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, são &lt;!-- MATH\n             $u = \\frac{\\pi}{2} + k\\pi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img222.svg\" alt=\"$u = \\frac{\\pi}{2} + k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, com\n            &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. O domínio da função tangente é então &lt;!-- MATH\n             $\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P97\" title=\"1P97\"&gt;\n            Como seno e cosseno são funções periódicas em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;, então a função tangente também será periódica. O que ocorre é que\n            o período da fração diminui para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt; pelo jogo de sinal entre numerador e denominador. De fato, as funções seno e\n            cosseno em módulo são periódicas de período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P98\" title=\"1P98\"&gt;\n            Vamos analizar os limites (laterais) nos pontos onde a função tangente não está definida. São os pontos &lt;!-- MATH\n             $u =\n            \\frac{\\pi}{2} + k\\pi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img222.svg\" alt=\"$u = \\frac{\\pi}{2} + k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. Pela periodicidade da função, basta analizar os limites em um destes pontose a análise valerá\n            para os demais. Vamos considerar, por simplicidade, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img226.svg\" alt=\"$k = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;!-- MATH\n             $\\frac{\\pi}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; teremos o\n            denominador &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; indo para &lt;span class=\"MATH\"&gt;0&lt;/span&gt;, e o numerador &lt;!-- MATH\n             ${\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; indo para 1 e, portanto, a razão vai para o infinito. Resta o\n            estudo do sinal. Se &lt;!-- MATH\n             $u \\to \\frac{\\pi}{2}^{+}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.74ex; vertical-align: -0.83ex; \" src=\"img/img229.svg\" alt=\"$u \\to \\frac{\\pi}{2}^{+}$\" loading=\"lazy\"&gt;&lt;/span&gt; então os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; são ligeiramente maiores que &lt;!-- MATH\n             $\\frac{\\pi}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            Neste caso, o seno será positivo e o cosseno negativo e, portanto,\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to \\frac{\\pi}{2}^{+}} {\\mathrm  {tg}}u = -\\infty.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P99\" title=\"1P99\"&gt;\n            &lt;img style=\"height: 3.97ex; vertical-align: -2.66ex; \" src=\"img/img230.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\frac{\\pi}{2}^{+}} {\\mathrm {tg}}u = -\\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P100\" title=\"1P100\"&gt;\n            Quando &lt;!-- MATH\n             $u \\to \\frac{\\pi}{2}^{-}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.74ex; vertical-align: -0.83ex; \" src=\"img/img231.svg\" alt=\"$u \\to \\frac{\\pi}{2}^{-}$\" loading=\"lazy\"&gt;&lt;/span&gt; então os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; serão menores que &lt;!-- MATH\n             $\\frac{\\pi}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; e, neste caso, seno e\n            cosseno serão positivos e, portanto,\n            &lt;/p&gt;\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to \\frac{\\pi}{2}^{-}} {\\mathrm  {tg}}u = \\infty.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P101\" title=\"1P101\"&gt;\n            &lt;img style=\"height: 3.73ex; vertical-align: -2.66ex; \" src=\"img/img232.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\frac{\\pi}{2}^{-}} {\\mathrm {tg}}u = \\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P102\" title=\"1P102\"&gt;\n            Estendendo esta análise para os outros valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos para todo &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n             &lt;/p&gt;\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to (\\frac{\\pi}{2}+k\\pi)^{+}} {\\mathrm  {tg}}u = - \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to (\\frac{\\pi}{2}+k\\pi)^{-}} {\\mathrm  {tg}}u = \\infty.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P103\" title=\"1P103\"&gt;\n            &lt;img style=\"height: 3.77ex; vertical-align: -2.46ex; \" src=\"img/img233.svg\" alt=\"$\\displaystyle \\lim_{u \\to (\\frac{\\pi}{2}+k\\pi)^{+}} {\\mathrm {tg}}u = - \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.53ex; vertical-align: -2.46ex; \" src=\"img/img234.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to (\\frac{\\pi}{2}+k\\pi)^{-}} {\\mathrm {tg}}u = \\infty.$\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P104\" title=\"1P104\"&gt;\n            Por se tratar de um quociente de duas funções, a função tangente será uma função contínua nos pontos em que o\n            denominador não se anulae cruzará o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; nos pontos em que o numerador se anular, isto é, para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img216.svg\" alt=\"$u = k \\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n             $k\n            \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Seu gráfico é como na figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"499\"&gt;&lt;/a&gt;\n            &lt;table id=\"1I13\" title=\"1I13\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.13:&lt;/strong&gt;\n            Gráfico da função tangente circular.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/./ftg.png\" alt=\"Image ftg\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P105\" title=\"1P105\"&gt;\n            É uma função ímpar. Não é injetora, mas é sobrejetora de &lt;!-- MATH\n             $\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Observe\n            atentamente que se analisada em apenas um dos intervalos de amplitude &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt;, da forma\n            &lt;!-- MATH\n             $(k\\pi-\\frac{\\pi}{2},k\\pi+\\frac{\\pi}{2})$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img235.svg\" alt=\"$(k\\pi-\\frac{\\pi}{2},k\\pi+\\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt; então torna-se uma função crescente e injetiva e, portanto, bijetiva.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P106\" title=\"1P106\"&gt;\n            A análise da função &lt;a name=\"507\"&gt;&lt;/a&gt; &lt;!-- MATH\n             $f(u) = {\\mathrm {ctg}}u = \\frac{\\cos u}{{\\mathrm {sen}}u}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.52ex; vertical-align: -0.85ex; \" src=\"img/img236.svg\" alt=\"$f(u) = {\\mathrm {ctg}}u = \\frac{\\cos u}{{\\mathrm {sen}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt; é feita da mesma forma que a\n            função tangente. Por se tratar de um quociente, o domínio de definição consiste dos valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; para os quais &lt;!-- MATH\n             ${\\mathrm {sen}}\n            u \\neq 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img237.svg\" alt=\"${\\mathrm {sen}}\n            u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. A função seno se anula nos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img216.svg\" alt=\"$u = k \\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta forma, o domínio da função cotangente é\n            o conjunto &lt;!-- MATH\n             $\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também esta função se anula nos pontos em que o numerador &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; se anula,\n            isto é, em &lt;!-- MATH\n             $u = \\frac{\\pi}{2} + k\\pi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img222.svg\" alt=\"$u = \\frac{\\pi}{2} + k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P107\" title=\"1P107\"&gt;\n            É também uma função periódica de período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt;. Vamos analisar os limites nos pontos onde esta função não está\n            definida, isto é, nos pontos onde o denominador se anula. Pela periodicidade, basta analizar os limites em um destes\n            pontos e esta análise valerá para os demais. Consideremos então por simplicidade o caso em que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img226.svg\" alt=\"$k = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, o ponto\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;0&lt;/span&gt; o denominador &lt;!-- MATH\n             ${\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;0&lt;/span&gt; também e o numerado se aproxima de 1.\n            A fração vai portanto para o infinito. Resta o estudo de sinais.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P108\" title=\"1P108\"&gt;\n            Quando &lt;!-- MATH\n             $u \\to 0^{+}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.13ex; \" src=\"img/img240.svg\" alt=\"$u \\to 0^{+}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é positivo; e tanto seno quanto cosseno são positivos, resultando\n             &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to 0^{+}} {\\mathrm {ctg}}u = \\infty,\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P109\" title=\"1P109\"&gt;\n            &lt;img style=\"height: 2.90ex; vertical-align: -1.82ex; \" src=\"img/img241.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} {\\mathrm {ctg}}u = \\infty, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P110\" style=\"text-indent: 0px !important;\" title=\"1P110\"&gt;\n            e quando &lt;!-- MATH\n             $u \\to 0^{-}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.13ex; \" src=\"img/img242.svg\" alt=\"$u \\to 0^{-}$\" loading=\"lazy\"&gt;&lt;/span&gt; então o seno será negativo e o cosseno positivo, neste caso\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to 0^{-}} {\\mathrm {ctg}}u = -\\infty.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P111\" title=\"1P111\"&gt;\n            &lt;img style=\"height: 3.14ex; vertical-align: -1.82ex; \" src=\"img/img243.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{-}} {\\mathrm {ctg}}u = -\\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P112\" title=\"1P112\"&gt;\n            Transmitindo estes fatos para os demais pontos onde a função não está definida, temos que\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to (k\\pi)^{+}} {\\mathrm {ctg}}u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to (k\\pi)^{-}} {\\mathrm {ctg}}u = -\\infty,\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P113\" title=\"1P113\"&gt;\n            &lt;img style=\"height: 3.31ex; vertical-align: -2.23ex; \" src=\"img/img244.svg\" alt=\"$\\displaystyle \\lim_{u \\to (k\\pi)^{+}} {\\mathrm {ctg}}u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.55ex; vertical-align: -2.23ex; \" src=\"img/img245.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to (k\\pi)^{-}} {\\mathrm {ctg}}u = -\\infty, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P114\" title=\"1P114\"&gt;\n            para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P115\" title=\"1P115\"&gt;\n            O gráfico então é a curva da figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"521\"&gt;&lt;/a&gt;\n            &lt;table id=\"1I14\" title=\"1I14\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.14:&lt;/strong&gt;\n            Gráfico da função cotangente circular.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/./fctg.png\" alt=\"Image fctg\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P116\" title=\"1P116\"&gt;\n            É uma função ímpar, sobrejetora mas não injetora de &lt;!-- MATH\n             $\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se analisada por partes,\n            isto é, em apenas um dos intervalos de amplitude &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt; da forma &lt;!-- MATH\n             $(k\\pi, (k+1)\\pi)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img246.svg\" alt=\"$(k\\pi, (k+1)\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então temos injetividade (e\n            portanto bijetividade) em qualquer um destes intervalos.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P117\" title=\"1P117\"&gt;\n            Agora a função &lt;a name=\"523\"&gt;&lt;/a&gt; &lt;!-- MATH\n             $f(u) = \\sec u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img247.svg\" alt=\"$f(u) = \\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Usando a identidade &lt;!-- MATH\n             $\\sec u = \\frac{1}{\\cos u}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img248.svg\" alt=\"$\\sec u = \\frac{1}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, vemos que o\n            domínio desta função fica caracterizado pelo conjunto &lt;!-- MATH\n             $\\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;, dos pontos tais que o\n            denominador não se anule. É uma função que nunca se anula, pois o numerador é fixo e não nulo. Note que o denominador\n            assume todos os valores reais entre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; (inclusive estes dois). Isto significa que a fração poderá resultar em\n            qualquer um dos valores maiores que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; ou menores que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; (inclusive estes dois). O conjunto imagem então é o\n            conjunto &lt;!-- MATH\n             $(-\\infty,-1] \\cup [1,\\infty)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P118\" title=\"1P118\"&gt;\n            Vamos verificar o comportamento da função nas proximidades dos pontos onde não estiver definida. Sendo o denominador\n            uma função periódica de período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; e o numerador constante, então o quociente é também uma função periódica de\n            período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. Por este motivo, olhemos para o intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img252.svg\" alt=\"$[0,2\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;; e usando a periodicidade deduzimos o\n            comportamento da função para os demais pontos onde não estiver definida.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P119\" title=\"1P119\"&gt;\n            Quando &lt;!-- MATH\n             $u \\to \\frac{\\pi}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img253.svg\" alt=\"$u \\to \\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; pela direita, os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximam de 0 negativamente e, portanto, &lt;!-- MATH\n             $f(u) \\to\n            -\\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img254.svg\" alt=\"$f(u) \\to\n            -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Pela esquerda, os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; vão para 0 positivamente e, portanto, &lt;!-- MATH\n             $f(u) \\to \\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img255.svg\" alt=\"$f(u) \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Resumindo,\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to (\\frac{\\pi}{2}+2k\\pi)^{+}} \\sec u\n             =-\\infty \\qquad \\text{e} \\qquad \\lim_{u \\to (\\frac{\\pi}{2}+2k\\pi)^{-}} \\sec u = \\infty. \n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P120\" title=\"1P120\"&gt;\n            &lt;img style=\"height: 3.77ex; vertical-align: -2.46ex; \" src=\"img/img256.svg\" alt=\"$\\displaystyle \\lim_{u \\to (\\frac{\\pi}{2}+2k\\pi)^{+}} \\sec u\n            =-\\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.53ex; vertical-align: -2.46ex; \" src=\"img/img257.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to (\\frac{\\pi}{2}+2k\\pi)^{-}} \\sec u = \\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P121\" title=\"1P121\"&gt;\n            Se &lt;!-- MATH\n             $u \\to \\frac{3\\pi}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img258.svg\" alt=\"$u \\to \\frac{3\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; pela direita, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de 0 positivamente e, portanto, &lt;!-- MATH\n             $f(u) \\to \\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img255.svg\" alt=\"$f(u) \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n            se &lt;!-- MATH\n             $u \\to \\frac{3\\pi}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img258.svg\" alt=\"$u \\to \\frac{3\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; pela esquerda, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de 0 negativamente e, portanto, &lt;!-- MATH\n             $f(u) \\to -\\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img254.svg\" alt=\"$f(u) \\to\n            -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            Isto é,\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to (\\frac{3\\pi}{2}+2k\\pi)^{+}} \\sec u\n             = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to (\\frac{3\\pi}{2}+2k\\pi)^{-}} \\sec u = -\\infty. \n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P122\" title=\"1P122\"&gt;\n            &lt;img style=\"height: 3.80ex; vertical-align: -2.73ex; \" src=\"img/img259.svg\" alt=\"$\\displaystyle \\lim_{u \\to (\\frac{3\\pi}{2}+2k\\pi)^{+}} \\sec u\n            = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.04ex; vertical-align: -2.73ex; \" src=\"img/img260.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to (\\frac{3\\pi}{2}+2k\\pi)^{-}} \\sec u = -\\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P123\" title=\"1P123\"&gt;\n            Note ainda que nos pontos &lt;!-- MATH\n             $\\{2k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img261.svg\" alt=\"$\\{2k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos &lt;!-- MATH\n             $\\cos u = 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img262.svg\" alt=\"$\\cos u = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; e então, &lt;!-- MATH\n             $\\sec u = 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img263.svg\" alt=\"$\\sec u = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; nestes pontos. Nos pontos\n            &lt;!-- MATH\n             $\\{(2k+1)\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img264.svg\" alt=\"$\\{(2k+1)\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; temos &lt;!-- MATH\n             $\\cos u = -1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img265.svg\" alt=\"$\\cos u = -1$\" loading=\"lazy\"&gt;&lt;/span&gt; e então &lt;!-- MATH\n             $\\sec u = -1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img266.svg\" alt=\"$\\sec u = -1$\" loading=\"lazy\"&gt;&lt;/span&gt; nestes pontos. O gráfico desta função é mostrado\n            na figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"550\"&gt;&lt;/a&gt;\n            &lt;table id=\"1I15\" title=\"1I15\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.15:&lt;/strong&gt;\n            Gráfico da função secante circular.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/./fsec.png\" alt=\"Image fsec\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P124\" title=\"1P124\"&gt;\n            Note que não é uma função injetora, nem sobrejetora de &lt;!-- MATH\n             $\\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P125\" title=\"1P125\"&gt;\n            Finalmente para a função &lt;a name=\"554\"&gt;&lt;/a&gt; &lt;!-- MATH\n             $f(u) = \\csc u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img267.svg\" alt=\"$f(u) = \\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;, levamos em conta que &lt;!-- MATH\n             $\\csc u = \\frac{1}{{\\mathrm {sen}}u}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img268.svg\" alt=\"$\\csc u = \\frac{1}{{\\mathrm {sen}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n            analisamos este quociente. O domínio é o conjunto de pontos tais que o denominador não se anula, ou seja, o conjunto\n            &lt;!-- MATH\n             $\\mathbb{R}- \\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também o denominador assume todos os valores (não nulos) entre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, a\n            função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img218.svg\" alt=\"$f(u)$\" loading=\"lazy\"&gt;&lt;/span&gt; assume todos os valores menores que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; e maiores que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt;. O conjunto imagem é então o conjunto\n            &lt;!-- MATH\n             $(-\\infty,-1] \\cup [1, \\infty)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P126\" title=\"1P126\"&gt;\n            Trata-se de um quociente com denominador periódico e numerador constante e, portanto, essa função também é periódica, de\n            período &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img41.svg\" alt=\"$2\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;. Basta analizarmos o intervalo &lt;!-- MATH\n             $[-\\frac{\\pi}{2},\\frac{3\\pi}{2}]$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img269.svg\" alt=\"$[-\\frac{\\pi}{2},\\frac{3\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt; e repetirmos o comportamento para os\n            demais pontos, onde o denominador se anula.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P127\" title=\"1P127\"&gt;\n            Quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; pela direita, então os valores de &lt;!-- MATH\n             ${\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximam de 0 positivamente e então &lt;!-- MATH\n             $f(u) = \\csc u \\to\n            \\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img271.svg\" alt=\"$f(u) = \\csc u \\to\n            \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; pela esquerda, então os valores de &lt;!-- MATH\n             ${\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; vão para 0 negativamente e então &lt;!-- MATH\n             $f(u) \\to -\\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img254.svg\" alt=\"$f(u) \\to\n            -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            Resumindo isto, temos\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to 2k\\pi^{+}} \\csc u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to 2k\\pi^{-}} \\csc u = -\\infty.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P128\" title=\"1P128\"&gt;\n            &lt;img style=\"height: 2.88ex; vertical-align: -1.80ex; \" src=\"img/img272.svg\" alt=\"$\\displaystyle \\lim_{u \\to 2k\\pi^{+}} \\csc u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.12ex; vertical-align: -1.80ex; \" src=\"img/img273.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 2k\\pi^{-}} \\csc u = -\\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P129\" title=\"1P129\"&gt;\n            Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt; pela direita, então o denominador &lt;!-- MATH\n             ${\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de 0 negativamente e então &lt;!-- MATH\n             $f(u)\n            \\to -\\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img254.svg\" alt=\"$f(u) \\to\n            -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt; pela esquerda, então o denominador se aproxima de 0 positivamente e, assim,\n            &lt;!-- MATH\n             $f(u) \\to \\infty$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img255.svg\" alt=\"$f(u) \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Dessa forma, temos\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{u \\to (2k+1)\\pi^{+}} \\csc u = -\\infty \\qquad \\text{e} \\qquad \\lim_{u \\to (2k+1)\\pi^{-}} \\csc u = \\infty.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"1P130\" title=\"1P130\"&gt;\n            &lt;img style=\"height: 3.55ex; vertical-align: -2.23ex; \" src=\"img/img274.svg\" alt=\"$\\displaystyle \\lim_{u \\to (2k+1)\\pi^{+}} \\csc u = -\\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.31ex; vertical-align: -2.23ex; \" src=\"img/img275.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to (2k+1)\\pi^{-}} \\csc u = \\infty. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P131\" title=\"1P131\"&gt;\n            Temos ainda que nos pontos &lt;!-- MATH\n             $\\{ \\frac{\\pi}{2} + 2k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img276.svg\" alt=\"$\\{ \\frac{\\pi}{2} + 2k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; o denominador da fração assume o valor 1 e,\n            portanto, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual a 1 nestes pontos. Analogamente nos pontos &lt;!-- MATH\n             $\\{ \\frac{3\\pi}{2} + 2k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img278.svg\" alt=\"$\\{ \\frac{3\\pi}{2} + 2k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; o\n            denominador é igual a -1 e, portanto, a função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt; assume o valor -1 nestes pontos. O gráfico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt; é representado na\n            figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"573\"&gt;&lt;/a&gt;\n            &lt;table id=\"1I16\" title=\"1I16\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.16:&lt;/strong&gt;\n            Gráfico da função cossecante circular.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/./fcossec.png\" alt=\"Image fcossec\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P132\" title=\"1P132\"&gt;\n            Não é uma função injetora e nem sobrejetora de &lt;!-- MATH\n             $\\mathbb{R}- \\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"1P133\" title=\"1P133\"&gt;\n            &lt;b&gt;Nota&lt;/b&gt;: Observe que a função seno é um deslocamento da função cosseno (e vice-versa) e por este motivo, os quocientes &lt;!-- MATH\n             $\\sec u =\n            \\frac{1}{\\cos u}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img248.svg\" alt=\"$\\sec u = \\frac{1}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n             $\\csc u = \\frac{1}{{\\mathrm {sen}}u}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img268.svg\" alt=\"$\\csc u = \\frac{1}{{\\mathrm {sen}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt; são também deslocamentos um do outro. Compare isto nos limites que\n            deduzimos e nos gráficos das duas funções.\n            &lt;!-- MATH\n             $\\blacksquare$\n             --&gt;\n            &lt;span style=\"float: right;\" class=\"MATH\"&gt;&lt;img style=\"height: 1.61ex; vertical-align: -0.10ex; \" src=\"img/img279.svg\" alt=\"$\\blacksquare$\" loading=\"lazy\"&gt;&lt;/span&gt;\n             &lt;/p&gt;\n            \n            &lt;p class=\" unidade\" id=\"1P134\" title=\"1P134\"&gt;\n            Vamos resumir em uma tabela o domínio e a imagem de cada uma das funções trigonométricas.\n            &lt;/p&gt;\n            \n            &lt;a id=\"590\"&gt;&lt;/a&gt;\n            &lt;table class=\"PAD  BORDER\" id=\"1T1\" title=\"1T1\"&gt;\n            &lt;caption style=\"margin-top: 15px;\"&gt;&lt;strong&gt;Tabela 1.1:&lt;/strong&gt;\n            Domínio e imagem das funções trigonométricas circulares&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"CENTER\"&gt;função&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;imagem&lt;/td&gt;\n            &lt;/tr&gt;\n           \n            &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             ${\\mathrm {sen}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;/tr&gt;\n            &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;/tr&gt;\n            &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             ${\\mathrm  {tg}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img281.svg\" alt=\"${\\mathrm {tg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;/tr&gt;\n            &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             ${\\mathrm {ctg}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img115.svg\" alt=\"${\\mathrm {ctg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;/tr&gt;\n            &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img121.svg\" alt=\"$\\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $(-\\infty,-1] \\cup [1,\\infty)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;/tr&gt;\n            &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img123.svg\" alt=\"$\\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $\\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n             $(-\\infty,-1] \\cup [1,\\infty)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;/tr&gt;\n          \n         \n            &lt;/tbody&gt;&lt;/table&gt;",
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.3 Continuidade das funções trigonométricas circulares",
    "text": "1.3 Continuidade das funções trigonométricas circulares\n\n                &lt;p class=\" unidade\" id=\"1P135\" title=\"1P135\"&gt;\n                Nesta seção vamos mostrar que as funções trigonométricas circulares são contínuas nos seus domínios de definição. Mais\n                precisamente, queremos primeiro mostrar que\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to a} {\\mathrm {sen}}u = {\\mathrm {sen}}a \\qquad \\text{e} \\qquad \\lim_{u \\to a} \\cos u = \\cos a,\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P136\" title=\"1P136\"&gt;\n                &lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img282.svg\" alt=\"$\\displaystyle \\lim_{u \\to a} {\\mathrm {sen}}u = {\\mathrm {sen}}a$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img283.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to a} \\cos u = \\cos a, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P137\" title=\"1P137\"&gt;\n                para qualquer &lt;!-- MATH\n                 $a \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para as outras quatro funções trigonométricas circulares, a continuidade seguirá da\n                continuidade destas duas funções e da propriedade de continuidade do quociente de funções contínuas.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"1P138\" title=\"1P138\"&gt;\n                Observando os gráficos destas duas funções nas figuras &lt;a href=\"#figfsin\"&gt;1.11&lt;/a&gt; e &lt;a href=\"#figfcos\"&gt;1.12&lt;/a&gt;, vemos que são gráficos formados\n                por linhas contínuas e, do ponto de vista gráfico, as funções são contínuas satisfazendo portanto os limites acima.\n                Todavia, precisamos ser mais rigorosos.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"1P139\" title=\"1P139\"&gt;\n                Vamos primeiro listar alguns resultados a respeito de limites que iremos utilizar nesta seção. Não vamos demonstrar\n                aqui estes resultados, pois está fora do nosso interesse principal. O leitor interessado nestas demonstrações pode\n                consultar [&lt;a href=\"/trigonometria-hiperbolica/referencias#Leithold\"&gt;4&lt;/a&gt;, Leithold].\n                &lt;/p&gt;\n                \n                &lt;div class=\"unidade\" id=\"1Teo2\" title=\"1Teo2\"&gt;&lt;a id=\"teopop\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img285.svg\" alt=\"$g$\" loading=\"lazy\"&gt;&lt;/span&gt; são funções cujos limites existem quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img286.svg\" alt=\"$x \\to a$\" loading=\"lazy\"&gt;&lt;/span&gt;, então &lt;br&gt;\n                &lt;/i&gt;&lt;dl class=\"COMPACT\"&gt;\n                &lt;dt&gt;a)&lt;/dt&gt;\n                &lt;dd&gt;&lt;!-- MATH\n                 $\\lim\\limits_{x \\to a} [f(x) + g(x)] = \\lim\\limits_{x \\to a} f(x) + \\lim\\limits_{x \\to a} g(x)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img287.svg\" alt=\"$\\lim\\limits_{x \\to a} [f(x) + g(x)] = \\lim\\limits_{x \\to a} f(x) + \\lim\\limits_{x \\to a} g(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                &lt;/dd&gt;\n                &lt;dt&gt;b)&lt;/dt&gt;\n                &lt;dd&gt;&lt;!-- MATH\n                 $\\lim\\limits_{x \\to a} kf(x) = k\\lim\\limits_{x \\to a} f(x)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img288.svg\" alt=\"$\\lim\\limits_{x \\to a} kf(x) = k\\lim\\limits_{x \\to a} f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; para qualquer &lt;!-- MATH\n                 $k \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img289.svg\" alt=\"$k \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                &lt;/dd&gt;\n                &lt;dt&gt;c)&lt;/dt&gt;\n                &lt;dd&gt;&lt;!-- MATH\n                 $\\lim\\limits_{x \\to a} [f(x)g(x)] = \\left(\\lim\\limits_{x \\to a} f(x)\\right)\\left(\\lim\\limits_{x \\to a} g(x) \\right)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.12ex; vertical-align: -1.60ex; \" src=\"img/img290.svg\" alt=\"$\\lim\\limits_{x \\to a} [f(x)g(x)] = \\left(\\lim\\limits_{x \\to a} f(x)\\right)\\left(\\lim\\limits_{x \\to a} g(x) \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                &lt;/dd&gt;\n                &lt;dt&gt;d)&lt;/dt&gt;\n                &lt;dd&gt;&lt;!-- MATH\n                 $\\lim\\limits_{x \\to a} \\dfrac{f(x)}{g(x)} = \\dfrac{\\lim\\limits_{x \\to a} f(x)}{\\lim\\limits_{x \\to a} g(x)}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 7.04ex; vertical-align: -3.05ex; \" src=\"img/img291.svg\" alt=\"$\\lim\\limits_{x \\to a} \\dfrac{f(x)}{g(x)} = \\dfrac{\\lim\\limits_{x \\to a} f(x)}{\\lim\\limits_{x \\to a} g(x)}$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                desde que &lt;!-- MATH\n                 $\\lim\\limits_{x \\to a} g(x) \\neq 0$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img292.svg\" alt=\"$\\lim\\limits_{x \\to a} g(x) \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/dd&gt;\n                &lt;/dl&gt;&lt;/div&gt;\n                \n                \n                &lt;div class=\"unidade\" id=\"1Teo3\" title=\"1Teo3\"&gt;&lt;a id=\"sandwich\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; (Teorema do confronto, ou Teorema do sanduíche)  &nbsp; \n                &lt;i&gt;Sejam &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img285.svg\" alt=\"$g$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img293.svg\" alt=\"$h$\" loading=\"lazy\"&gt;&lt;/span&gt; funções tais que &lt;!-- MATH\n                 $f(x) \\leq g(x) \\leq h(x)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img294.svg\" alt=\"$f(x) \\leq g(x) \\leq h(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; em algum intervalo em torno de um ponto\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\" loading=\"lazy\"&gt;&lt;/span&gt;, exceto possivelmente no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{x \\to a} f(x) = L = \\lim_{x \\to a} h(x),\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P140\" title=\"1P140\"&gt;\n                &lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img296.svg\" alt=\"$\\displaystyle \\lim_{x \\to a} f(x) = L = \\lim_{x \\to a} h(x), $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                &lt;i&gt;\n                então\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{x \\to a} g(x) = L.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P141\" title=\"1P141\"&gt;\n                &lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img297.svg\" alt=\"$\\displaystyle \\lim_{x \\to a} g(x) = L. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                &lt;div class=\"unidade\" id=\"1Teo4\" title=\"1Teo4\"&gt;&lt;a id=\"teofgdif\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img285.svg\" alt=\"$g$\" loading=\"lazy\"&gt;&lt;/span&gt; são funções tais que &lt;!-- MATH\n                 $f(x) = g(x)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img298.svg\" alt=\"$f(x) = g(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; em algum intervalo em torno de um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\" loading=\"lazy\"&gt;&lt;/span&gt;, exceto\n                possivelmente no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\" loading=\"lazy\"&gt;&lt;/span&gt;, então o limite de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img286.svg\" alt=\"$x \\to a$\" loading=\"lazy\"&gt;&lt;/span&gt; existe se e somente se existe o limite de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img300.svg\" alt=\"$g(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img286.svg\" alt=\"$x \\to a$\" loading=\"lazy\"&gt;&lt;/span&gt;. Além disso,\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{x \\to a} f(x) = \\lim_{x \\to a} g(x).\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P142\" title=\"1P142\"&gt;\n                &lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img301.svg\" alt=\"$\\displaystyle \\lim_{x \\to a} f(x) = \\lim_{x \\to a} g(x). $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                &lt;div class=\"unidade\" id=\"1Teo5\" title=\"1Teo5\"&gt;&lt;a id=\"teomudv\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;O limite &lt;!-- MATH\n                 $\\lim\\limits_{x \\to 0} f(x+a)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img302.svg\" alt=\"$\\lim\\limits_{x \\to 0} f(x+a)$\" loading=\"lazy\"&gt;&lt;/span&gt; existe, se e somente se, existe o limite &lt;!-- MATH\n                 $\\lim\\limits_{x \\to a} f(x)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.27ex; vertical-align: -1.60ex; \" src=\"img/img303.svg\" alt=\"$\\lim\\limits_{x \\to a} f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;; e mais\n                ainda,\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{x \\to 0} f(x+a) = \\lim_{x \\to a} f(x).\n                \\end{displaymath}\n                 --&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"1P143\" title=\"1P143\"&gt;\n                &lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img304.svg\" alt=\"$\\displaystyle \\lim_{x \\to 0} f(x+a) = \\lim_{x \\to a} f(x). $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P144\" title=\"1P144\"&gt;\n                Agora começamos o trabalho de provar a continuidade das funções seno e cosseno.\n                \n                &lt;/p&gt;\n                \n                &lt;div class=\"unidade\" id=\"1Teo6\" title=\"1Teo6\"&gt;&lt;a id=\"limfundsin\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;O limite\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0} \\frac{{\\mathrm {sen}}u}{u}\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P145\" title=\"1P145\"&gt;\n                &lt;img style=\"height: 4.26ex; vertical-align: -1.78ex; \" src=\"img/img305.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\frac{{\\mathrm {sen}}u}{u} $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;i&gt;\n                existe e é igual a 1.\n                &lt;/i&gt;&lt;/div&gt;\n                \n                \n                &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                \n                Provaremos que os limites laterais existem e são iguais a 1. Consideremos primeiro o caso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img306.svg\" alt=\"$u&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;; e podemos também\n                considerar que &lt;!-- MATH\n                 $u \\in (0,\\frac{\\pi}{2})$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img307.svg\" alt=\"$u \\in (0,\\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;. No círculo trigonométrico, construímos o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; de comprimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então\n                o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img308.svg\" alt=\"$A\\hat{O}V$\" loading=\"lazy\"&gt;&lt;/span&gt; tem medida &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; radianos. Consideremos a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; de equação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img309.svg\" alt=\"$u=1$\" loading=\"lazy\"&gt;&lt;/span&gt;, perpendicular ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                passando pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img310.svg\" alt=\"$V$\" loading=\"lazy\"&gt;&lt;/span&gt;. Prolonguemos o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img100.svg\" alt=\"$OA$\" loading=\"lazy\"&gt;&lt;/span&gt; até interceptar a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; em um ponto que designaremos por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                Consideremos então o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt;, o setor circular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; e o triângulo retângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img312.svg\" alt=\"$OVP$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                \n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"650\"&gt;&lt;/a&gt;\n                &lt;table id=\"1I17\" title=\"1I17\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.17:&lt;/strong&gt;\n                Visualização geométrica do limite.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/./limfund.png\" alt=\"Image limfund\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P146\" title=\"1P146\"&gt;\n                Vemos claramente que a área do triângulo retângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img312.svg\" alt=\"$OVP$\" loading=\"lazy\"&gt;&lt;/span&gt; é maior que a área do setor circular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; que por sua vez é\n                maior que a área do triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt;. O triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img312.svg\" alt=\"$OVP$\" loading=\"lazy\"&gt;&lt;/span&gt; tem base com medida 1 e altura com medida &lt;!-- MATH\n                 ${\\mathrm  {tg}}u = \\frac{{\\mathrm {sen}}\n                u}{\\cos u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.47ex; vertical-align: -0.85ex; \" src=\"img/img114.svg\" alt=\"${\\mathrm {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, a sua área é &lt;!-- MATH\n                 $\\frac{{\\mathrm {sen}}u}{2\\cos u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.47ex; vertical-align: -0.85ex; \" src=\"img/img313.svg\" alt=\"$\\frac{{\\mathrm {sen}}u}{2\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt;. O setor circular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; tem área igual a &lt;!-- MATH\n                 $\\frac{u}{2}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img61.svg\" alt=\"$\\frac{u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. O\n                triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; tem base com medida 1 e altura com medida &lt;!-- MATH\n                 ${\\mathrm {sen}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, área igual a &lt;!-- MATH\n                 $\\frac{{\\mathrm {sen}}u}{2}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img314.svg\" alt=\"$\\frac{{\\mathrm {sen}}u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Nestes\n                termos\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\frac{{\\mathrm {sen}}u}{2} &lt; \\frac{u}{2} &lt; \\frac{{\\mathrm {sen}}u}{2\\cos u}.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P147\" title=\"1P147\"&gt;\n                &lt;img style=\"height: 4.06ex; vertical-align: -1.57ex; \" src=\"img/img315.svg\" alt=\"$\\displaystyle \\frac{{\\mathrm {sen}}u}{2} &lt; \\frac{u}{2} &lt; \\frac{{\\mathrm {sen}}u}{2\\cos u}. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P148\" title=\"1P148\"&gt;\n                Multiplicando tudo por 2 e dividindo por &lt;!-- MATH\n                 ${\\mathrm {sen}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; (que é positivo), temos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                1 &lt; \\frac{u}{{\\mathrm {sen}}u} &lt; \\frac{1}{\\cos u},\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"1P149\" title=\"1P149\"&gt;\n                &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img316.svg\" alt=\"$\\displaystyle 1 &lt; \\frac{u}{{\\mathrm {sen}}u} &lt; \\frac{1}{\\cos u}, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P150\" title=\"1P150\"&gt;\n                ou ainda\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                1 &gt; \\frac{{\\mathrm {sen}}u}{u} &gt; \\cos u.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P151\" title=\"1P151\"&gt;\n                &lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img317.svg\" alt=\"$\\displaystyle 1 &gt; \\frac{{\\mathrm {sen}}u}{u} &gt; \\cos u. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P152\" title=\"1P152\"&gt;\n                Da primeira desigualdade, temos que &lt;!-- MATH\n                 ${\\mathrm {sen}}u &lt; u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.33ex; vertical-align: -0.14ex; \" src=\"img/img318.svg\" alt=\"${\\mathrm {sen}}u &lt; u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Substituindo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; por &lt;!-- MATH\n                 $\\frac{1}{2}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img319.svg\" alt=\"$\\frac{1}{2}u$\" loading=\"lazy\"&gt;&lt;/span&gt; temos que &lt;!-- MATH\n                 ${\\mathrm {sen}}(\\frac{1}{2}u) &lt;\n                \\frac{1}{2}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img320.svg\" alt=\"${\\mathrm {sen}}(\\frac{1}{2}u) &lt;\n                \\frac{1}{2}u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim,\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"1P153\" title=\"1P153\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img321.svg\" alt=\"$\\displaystyle \\frac{1-\\cos u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.08ex; vertical-align: -1.55ex; \" src=\"img/img322.svg\" alt=\"$\\displaystyle = \\frac{1-\\cos(\\frac{1}{2}u+\\frac{1}{2}u)}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.08ex; vertical-align: -1.55ex; \" src=\"img/img323.svg\" alt=\"$\\displaystyle = \\frac{1-\\cos^{2}(\\frac{1}{2}u)+{\\mathrm {sen}}^{2}(\\frac{1}{2}u)}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.08ex; vertical-align: -1.55ex; \" src=\"img/img324.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}^{2}(\\frac{1}{2}u)+{\\mathrm {sen}}^{2}(\\frac{1}{2}u)}{2} = {\\mathrm {sen}}^{2}(\\tfrac{1}{2}u) &lt; (\\tfrac{1}{2}u)^{2}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"1P154\" title=\"1P154\"&gt;\n                Segue que &lt;!-- MATH\n                 $\\frac{1-\\cos u}{2} &lt; (\\tfrac{1}{2}u)^{2}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img325.svg\" alt=\"$\\frac{1-\\cos u}{2} &lt; (\\tfrac{1}{2}u)^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; e reorganizando os termos obtemos &lt;!-- MATH\n                 $\\cos u &gt; 1-\\frac{1}{2}u^{2}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img326.svg\" alt=\"$\\cos u &gt; 1-\\frac{1}{2}u^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                Logo\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                1 &gt; \\frac{{\\mathrm {sen}}u}{u} &gt; \\cos u &gt; 1-\\frac{1}{2}u^{2}.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P155\" title=\"1P155\"&gt;\n                &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img327.svg\" alt=\"$\\displaystyle 1 &gt; \\frac{{\\mathrm {sen}}u}{u} &gt; \\cos u &gt; 1-\\frac{1}{2}u^{2}. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P156\" title=\"1P156\"&gt;\n                Como o limite das funções 1 e &lt;!-- MATH\n                 $(1-\\frac{1}{2}u^{2})$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img328.svg\" alt=\"$(1-\\frac{1}{2}u^{2})$\" loading=\"lazy\"&gt;&lt;/span&gt; existem e são iguais a 1 quando &lt;!-- MATH\n                 $u \\to 0^{+}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.13ex; \" src=\"img/img240.svg\" alt=\"$u \\to 0^{+}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então segue do\n                Teorema do confronto (Teorema &lt;a href=\"#sandwich\"&gt;1.3&lt;/a&gt;) que\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0^{+}} \\frac{{\\mathrm {sen}}u}{u} = 1.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P157\" title=\"1P157\"&gt;\n                &lt;img style=\"height: 4.31ex; vertical-align: -1.82ex; \" src=\"img/img329.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} \\frac{{\\mathrm {sen}}u}{u} = 1. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P158\" title=\"1P158\"&gt;\n                Agora, se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img52.svg\" alt=\"$u &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, levando em conta que &lt;!-- MATH\n                 $\\frac{{\\mathrm {sen}}u}{u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img330.svg\" alt=\"$\\frac{{\\mathrm {sen}}u}{u}$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função par, então o comportamento pela direita de zero é\n                o mesmo pela esquerda. Segue que\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0^{-}} \\frac{{\\mathrm {sen}}u}{u} = 1,\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P159\" title=\"1P159\"&gt;\n                &lt;img style=\"height: 4.31ex; vertical-align: -1.82ex; \" src=\"img/img331.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{-}} \\frac{{\\mathrm {sen}}u}{u} = 1,$\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                também e isso prova o limite desejado.\n                &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                &lt;/div&gt;\n                &lt;div class=\"unidade\" id=\"1Teo7\" title=\"1Teo7\"&gt;&lt;a id=\"teorema1-7\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;As funções seno e cosseno são contínuas em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0} \\cos u = 1 \\qquad \\text{e} \\qquad \\lim_{u \\to 0} {\\mathrm {sen}}u = 0.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P160\" title=\"1P160\"&gt;\n                &lt;img style=\"height: 3.33ex; vertical-align: -1.78ex; \" src=\"img/img332.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cos u = 1$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.33ex; vertical-align: -1.78ex; \" src=\"img/img333.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 0} {\\mathrm {sen}}u = 0.$\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                \n                &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                \n                Para o primeiro limite, supondo primeiro &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img306.svg\" alt=\"$u&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;, usamos a desigualdade\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                1 &gt; \\frac{{\\mathrm {sen}}u}{u} &gt; \\cos u &gt; 1-\\frac{1}{2}u^{2},\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P161\" title=\"1P161\"&gt;\n                &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img334.svg\" alt=\"$\\displaystyle 1 &gt; \\frac{{\\mathrm {sen}}u}{u} &gt; \\cos u &gt; 1-\\frac{1}{2}u^{2}, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                &lt;p class=\" unidade\" id=\"1P162\" style=\"text-indent: 0px !important;\" title=\"1P162\"&gt;\n                obtida na demonstração do teorema anterior; e o teorema do confronto garante que &lt;!-- MATH\n                 $\\lim\\limits_{u \\to 0^{+}} \\cos u =\n                1$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img335.svg\" alt=\"$\\lim\\limits_{u \\to 0^{+}} \\cos u =\n                1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img52.svg\" alt=\"$u &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt; lembremos que cosseno é uma função par e então o comportamento à esquerda de 0 é o mesmo comportamento\n                à direita de 0. Segue que\n                &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0^{-}} \\cos u = \\lim_{u \\to 0^{+}} \\cos u = 1,\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P163\" title=\"1P163\"&gt;\n                &lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img336.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{-}} \\cos u = \\lim_{u \\to 0^{+}} \\cos u = 1, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                e isto prova o primeiro limite.\n                &lt;p class=\" unidade\" id=\"1P164\" title=\"1P164\"&gt;&lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"1P165\" title=\"1P165\"&gt;\n                Para provar o segundo limite, usaremos o item (c) do teorema &lt;a href=\"#teopop\"&gt;1.2&lt;/a&gt;. Como os limites de &lt;!-- MATH\n                 $\\frac{{\\mathrm {sen}}u}{u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img330.svg\" alt=\"$\\frac{{\\mathrm {sen}}u}{u}$\" loading=\"lazy\"&gt;&lt;/span&gt; e de\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; existem quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; então o limite do produto existe e\n                 &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0} u \\frac{{\\mathrm {sen}}u}{u} = \\lim_{u \\to 0} u \\cdot \\lim_{u \\to 0} \\frac{{\\mathrm {sen}}u}{u} = 0 \\cdot 1 = 0.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P166\" title=\"1P166\"&gt;\n                &lt;img style=\"height: 4.26ex; vertical-align: -1.78ex; \" src=\"img/img337.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} u \\frac{{\\mathrm {sen}}u}{u} = \\lim_{u \\to 0} u \\cdot \\lim_{u \\to 0} \\frac{{\\mathrm {sen}}u}{u} = 0 \\cdot 1 = 0. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P167\" title=\"1P167\"&gt;\n                Agora, como &lt;!-- MATH\n                 $\\frac{u {\\mathrm {sen}}u}{u} = {\\mathrm {sen}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img338.svg\" alt=\"$\\frac{u {\\mathrm {sen}}u}{u} = {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img339.svg\" alt=\"$u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; então do teorema &lt;a href=\"#teofgdif\"&gt;1.4&lt;/a&gt; segue que\n                 &lt;/p&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0} {\\mathrm {sen}}u = \\lim_{u \\to 0} \\frac{u {\\mathrm {sen}}u}{u} = 0,\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P168\" title=\"1P168\"&gt;\n                &lt;img style=\"height: 4.26ex; vertical-align: -1.78ex; \" src=\"img/img340.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} {\\mathrm {sen}}u = \\lim_{u \\to 0} \\frac{u {\\mathrm {sen}}u}{u} = 0, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                e isto finaliza esta demonstração.\n                &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                &lt;/div&gt;\n                \n                \n                &lt;div class=\"unidade\" id=\"1Teo8\" title=\"1Teo8\"&gt;&lt;a id=\"teorema1-8\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;Para qualquer &lt;!-- MATH\n                 $a \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tem-se\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0} \\cos(u+a) = \\cos a \\qquad \\text{e} \\qquad \\lim_{u \\to 0} {\\mathrm {sen}}(u+a) = {\\mathrm {sen}}a.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P169\" title=\"1P169\"&gt;\n                &lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img341.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cos(u+a) = \\cos a$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img342.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 0} {\\mathrm {sen}}(u+a) = {\\mathrm {sen}}a.$\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                \n                Usando a identidade trigonométrica para a soma de arcos do cosseno,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\cos(u+a) = \\cos u \\cos a - {\\mathrm {sen}}u {\\mathrm {sen}}a,\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P170\" title=\"1P170\"&gt;\n                &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img343.svg\" alt=\"$\\displaystyle \\cos(u+a) = \\cos u \\cos a - {\\mathrm {sen}}u {\\mathrm {sen}}a, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                e como existem os limites de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; e de &lt;!-- MATH\n                 ${\\mathrm {sen}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, então existem os limites de &lt;!-- MATH\n                 $\\cos u \\cos a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img344.svg\" alt=\"$\\cos u \\cos a$\" loading=\"lazy\"&gt;&lt;/span&gt; e de\n                &lt;!-- MATH\n                 ${\\mathrm {sen}}u {\\mathrm {sen}}a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img345.svg\" alt=\"${\\mathrm {sen}}u {\\mathrm {sen}}a$\" loading=\"lazy\"&gt;&lt;/span&gt; e também existe o limite da soma destes dois termos, quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Segue que\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P171\" title=\"1P171\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img346.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cos(u+a)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img347.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} (\\cos u \\cos a - {\\mathrm {sen}}u {\\mathrm {sen}}a)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img348.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} (\\cos u \\cos a) - \\lim_{u \\to 0} ({\\mathrm {sen}}u {\\mathrm {sen}}a)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img349.svg\" alt=\"$\\displaystyle = (\\cos a) \\lim_{u \\to 0} \\cos u - ({\\mathrm {sen}}a) \\lim_{u \\to 0} {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img350.svg\" alt=\"$\\displaystyle = (\\cos a) \\cdot 1 - ({\\mathrm {sen}}a) \\cdot 0 = \\cos a.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"1P172\" title=\"1P172\"&gt;\n                Usando agora a identidade trigonométrica para a soma de arcos do seno e a existência dos limites de &lt;!-- MATH\n                 ${\\mathrm {sen}}u \\cos a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img351.svg\" alt=\"${\\mathrm {sen}}u \\cos a$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n                de &lt;!-- MATH\n                 ${\\mathrm {sen}}a \\cos u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img352.svg\" alt=\"${\\mathrm {sen}}a \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;, quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"1P173\" title=\"1P173\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img353.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} {\\mathrm {sen}}(u+a)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img354.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} ({\\mathrm {sen}}u \\cos a + {\\mathrm {sen}}a \\cos u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img355.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} ({\\mathrm {sen}}u \\cos a) + \\lim_{u \\to 0} ({\\mathrm {sen}}a \\cos u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img356.svg\" alt=\"$\\displaystyle = (\\cos a) \\lim_{u \\to 0} {\\mathrm {sen}}u + ({\\mathrm {sen}}a) \\lim_{u \\to 0} \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img357.svg\" alt=\"$\\displaystyle = (\\cos a) \\cdot 0 + ({\\mathrm {sen}}a) \\cdot 1 = {\\mathrm {sen}}a,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                e isso termina essa demonstração.\n                &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                &lt;/div&gt;\n                &lt;br /&gt;\n                &lt;p class=\" unidade\" id=\"1P174\" title=\"1P174\"&gt;\n                Os limites indicados no início desta seção seguem agora imediatamente do teorema &lt;a href=\"#teomudv\"&gt;1.5&lt;/a&gt; e deste último teorema.\n                &lt;/p&gt;\n                \n                &lt;div class=\"unidade\" id=\"1Teo9\" title=\"1Teo9\"&gt;&lt;a id=\"corolario\"&gt;&lt;b&gt;Corolário  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;9&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                &lt;i&gt;As funções seno e cosseno são contínuas em qualquer ponto &lt;!-- MATH\n                 $a \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to a} \\cos u = \\cos a \\qquad \\text{e} \\qquad \\lim_{u \\to a} {\\mathrm {sen}}u = {\\mathrm {sen}}a.\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"1P175\" title=\"1P175\"&gt;\n                &lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img358.svg\" alt=\"$\\displaystyle \\lim_{u \\to a} \\cos u = \\cos a$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img359.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to a} {\\mathrm {sen}}u = {\\mathrm {sen}}a.$\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"1P176\" title=\"1P176\"&gt;\n                Podemos agora analisar a continuidade das outras quatro funções trigonométricas circulares, já que estas são escritas\n                como um quociente em termos de seno e cosseno. Usando o item (d) do teorema &lt;a href=\"#teopop\"&gt;1.2&lt;/a&gt;, podemos facilmente provar as\n                próximas afirmações.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"1P177\" title=\"1P177\"&gt;\n                As funções tangente, cotangente, secante e cossecante são contínuas nos seus domínios de definição. De outra forma,\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;table class=\"PAD \"&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 $\\lim\\limits_{u \\to a} {\\mathrm  {tg}}u = {\\mathrm  {tg}}a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img360.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {tg}}u = {\\mathrm {tg}}a$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;desde que &nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                 $a \\notin \\{\\frac{\\pi}{2}+k\\pi, \\, k \\in \\mathbb{Z}\\}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img361.svg\" alt=\"$a \\notin \\{\\frac{\\pi}{2}+k\\pi, \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 $\\lim\\limits_{u \\to a} {\\mathrm {ctg}}u = {\\mathrm {ctg}}a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img362.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {ctg}}u = {\\mathrm {ctg}}a$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;desde que &nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                 $a \\notin \\{k\\pi, \\, k \\in \\mathbb{Z}\\}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img363.svg\" alt=\"$a \\notin \\{k\\pi, \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 $\\lim\\limits_{u \\to a} \\sec u = \\sec a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img364.svg\" alt=\"$\\lim\\limits_{u \\to a} \\sec u = \\sec a$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;desde que &nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                 $a \\notin \\{\\frac{\\pi}{2}+k\\pi, \\, k \\in \\mathbb{Z}\\}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img361.svg\" alt=\"$a \\notin \\{\\frac{\\pi}{2}+k\\pi, \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 $\\lim\\limits_{u \\to a} \\csc u = \\csc a$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.69ex; vertical-align: -1.60ex; \" src=\"img/img365.svg\" alt=\"$\\lim\\limits_{u \\to a} \\csc u = \\csc a$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;desde que &nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                 $a \\notin \\{k\\pi, \\, k \\in \\mathbb{Z}\\}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img363.svg\" alt=\"$a \\notin \\{k\\pi, \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;",
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.4 Derivadas de funções trigonométricas circulares",
    "text": "1.4 Derivadas de funções trigonométricas circulares\n\n                    &lt;p class=\" unidade\" id=\"1P178\" title=\"1P178\"&gt;\n                    Nesta seção estamos interessados em obter as fórmulas de derivada para as seis funções trigonométricas circulares. Para\n                    isto, usaremos a definição de derivada\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} f(u) = f'(u) = \\lim_{h \\to 0} \\frac{f(u+h) - f(u)}{h},\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P179\" title=\"1P179\"&gt;\n                    &lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img366.svg\" alt=\"$\\displaystyle \\frac{d}{du} f(u) = f'(u) = \\lim_{h \\to 0} \\frac{f(u+h) - f(u)}{h}, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    para obter as derivadas das funções seno e cosseno. Feito isto, usaremos a regra do quociente para determinar as\n                    derivadas das outras quatro funções trigonométricas. Lembremos rapidamente da regra do quociente para derivadas. Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                    e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img285.svg\" alt=\"$g$\" loading=\"lazy\"&gt;&lt;/span&gt; são funções deriváveis em um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                     $g(u) \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img367.svg\" alt=\"$g(u) \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, então o quociente é derivável neste ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;; e mais\n                    ainda\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} \\left( \\frac{f(u)}{g(u)} \\right) = \\frac{f'(u)g(u) - f(u)g'(u)}{[g(u)]^{2}}.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P180\" title=\"1P180\"&gt;\n                    &lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img368.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\left( \\frac{f(u)}{g(u)} \\right) = \\frac{f'(u)g(u) - f(u)g'(u)}{[g(u)]^{2}}. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"1P181\" title=\"1P181\"&gt;\n                    Primeiro precisamos estabelecer um limite necessário para a obtenção da derivada da função cosseno.\n                    \n                    &lt;/p&gt;\n                    \n                    &lt;div class=\"unidade\" id=\"1Teo10\" title=\"1Teo10\"&gt;&lt;a id=\"limfundcos\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;10&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;O limite\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{h \\to 0} \\frac{1 - \\cos h}{h}\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P182\" title=\"1P182\"&gt;\n                    &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img369.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{1 - \\cos h}{h} $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;i&gt;\n                    existe e é igual a 0.\n                    &lt;/i&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                    \n                    Podemos supor que &lt;!-- MATH\n                     $h \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img370.svg\" alt=\"$h \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;. Neste intevalo temos que &lt;!-- MATH\n                     $(1+\\cos h) \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img371.svg\" alt=\"$(1+\\cos h) \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e então\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P183\" title=\"1P183\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img372.svg\" alt=\"$\\displaystyle \\frac{1 - \\cos h}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img373.svg\" alt=\"$\\displaystyle = \\frac{(1 - \\cos h)(1+\\cos h)}{h(1+\\cos h)}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.42ex; vertical-align: -2.07ex; \" src=\"img/img374.svg\" alt=\"$\\displaystyle = \\frac{1-\\cos^{2}h}{h (1+\\cos h)} = \\frac{{\\mathrm {sen}}^{2}h}{h(1+\\cos h)} = \\frac{{\\mathrm {sen}}h}{h} \\cdot \\frac{{\\mathrm {sen}}h}{1+\\cos h}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P184\" title=\"1P184\"&gt;\n                    Como o limite da função &lt;!-- MATH\n                     $\\frac{{\\mathrm {sen}}h}{h}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img375.svg\" alt=\"$\\frac{{\\mathrm {sen}}h}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt; existe (ver proposição &lt;a href=\"#limfundsin\"&gt;1.6&lt;/a&gt;) e o limite da função &lt;!-- MATH\n                     $\\frac{{\\mathrm {sen}}\n                    h}{1+\\cos h}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img376.svg\" alt=\"$\\frac{{\\mathrm {sen}}\n                    h}{1+\\cos h}$\" loading=\"lazy\"&gt;&lt;/span&gt; existe como função contínua de &lt;!-- MATH\n                     $h \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img370.svg\" alt=\"$h \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;, então do item (c) do teorema\n                    &lt;a href=\"#teopop\"&gt;1.2&lt;/a&gt;, temos\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P185\" title=\"1P185\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img369.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{1 - \\cos h}{h} $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img377.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}h}{h} \\cdot \\frac{{\\mathrm {sen}}h}{1+\\cos h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img378.svg\" alt=\"$\\displaystyle = \\left( \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}h}{h} \\right) \\cdot ...\n                    ...\\mathrm {sen}}h}{1+\\cos h} \\right)\n                    = 1 \\cdot \\left( \\tfrac{0}{1+1} \\right) = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    e isto termina esta demonstração.\n                    &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"1P186\" title=\"1P186\"&gt;\n                    Agora estamos prontos para obter as derivadas das funções trigonométricas. Comecemos com &lt;a name=\"876\"&gt;&lt;/a&gt; &lt;!-- MATH\n                     $f(u) =\n                    {\\mathrm {sen}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img379.svg\" alt=\"$f(u) =\n                    {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida em toda a reta real. Aplicando a definição de derivada temos que\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} {\\mathrm {sen}}u = \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}(u+h)-{\\mathrm {sen}}u}{h}\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P187\" title=\"1P187\"&gt;\n                    &lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img380.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {sen}}u = \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}(u+h)-{\\mathrm {sen}}u}{h} $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    &lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P188\" title=\"1P188\"&gt;\n                    para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que o limite exista. Observe que para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; o limite existe e então a derivada da função seno\n                    existe em pelo menos um ponto. Vamos provar que o limite existe para qualquer &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"1P189\" title=\"1P189\"&gt;\n                    Desta forma, para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P190\" title=\"1P190\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img381.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}(u+h)-{\\mathrm {sen}}u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img382.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}u \\cos h + {\\mathrm {sen}}h \\cos u - {\\mathrm {sen}}u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img383.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\left( {\\mathrm {sen}}u \\frac{\\cos h - 1}{h} + \\frac{{\\mathrm {sen}}h}{h} \\cos u \\right).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P191\" title=\"1P191\"&gt;\n                    As duas parcelas dentro do último limite, são funções cujo limite em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img293.svg\" alt=\"$h$\" loading=\"lazy\"&gt;&lt;/span&gt; existe para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; (ver proposições\n                    &lt;a href=\"#limfundsin\"&gt;1.6&lt;/a&gt; e &lt;a href=\"#limfundcos\"&gt;1.10&lt;/a&gt;). Então\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P192\" title=\"1P192\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img381.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}(u+h)-{\\mathrm {sen}}u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img384.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} {\\mathrm {sen}}u \\frac{\\cos h - 1}{h} + \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}h}{h} \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img385.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}u \\left( \\lim_{h \\to 0} \\frac{\\cos h - 1}{h} \\right) + \\cos u \\left( \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}h}{h} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img386.svg\" alt=\"$\\displaystyle = 0 \\cdot {\\mathrm {sen}}u + 1 \\cdot \\cos u = \\cos u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P193\" title=\"1P193\"&gt;\n                    Como o limite existe para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, a função &lt;!-- MATH\n                     $f(u) = {\\mathrm {sen}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img379.svg\" alt=\"$f(u) =\n                    {\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é derivável em todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;; e, além disso,\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} {\\mathrm {sen}}u = \\cos u.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P194\" title=\"1P194\"&gt;\n                    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img387.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {sen}}u = \\cos u. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"1P195\" title=\"1P195\"&gt;\n                    Tomemos agora a função &lt;a name=\"916\"&gt;&lt;/a&gt; &lt;!-- MATH\n                     $f(u) = \\cos u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img388.svg\" alt=\"$f(u) = \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida em toda a reta real. Da definição de\n                    derivada,\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} \\cos u = \\lim_{h \\to 0} \\frac{\\cos(u+h) - \\cos u}{h}\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P196\" title=\"1P196\"&gt;\n                    &lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img389.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cos u = \\lim_{h \\to 0} \\frac{\\cos(u+h) - \\cos u}{h} $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P197\" title=\"1P197\"&gt;\n                    para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que o limite existe. Observe novamente que já provamos que este limite existe pelo menos para\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos que\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P198\" title=\"1P198\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img390.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{\\cos(u+h) - \\cos u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img391.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{\\cos u \\cos h - {\\mathrm {sen}}u {\\mathrm {sen}}h - \\cos u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img392.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\left( \\cos u \\frac{\\cos h - 1}{h} - {\\mathrm {sen}}u \\frac{{\\mathrm {sen}}h}{h} \\right).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P199\" title=\"1P199\"&gt;\n                    Novamente, as parcelas dentro do limite são funções tais que o limite existe em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img293.svg\" alt=\"$h$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P200\" title=\"1P200\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img390.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{\\cos(u+h) - \\cos u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img393.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\cos u \\frac{\\cos h - 1}{h} - \\lim_{h \\to 0} {\\mathrm {sen}}u \\frac{{\\mathrm {sen}}h}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img394.svg\" alt=\"$\\displaystyle = \\cos u \\left( \\lim_{h \\to 0} \\frac{\\cos h - 1}{h} \\right) - {\\mathrm {sen}}u \\left( \\lim_{h \\to 0} \\frac{{\\mathrm {sen}}h}{h} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img395.svg\" alt=\"$\\displaystyle = 0 \\cdot \\cos u - 1 \\cdot {\\mathrm {sen}}u = - {\\mathrm {sen}}u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P201\" title=\"1P201\"&gt;\n                    O limite existe, portanto, para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e, assim, segue que a função &lt;!-- MATH\n                     $f(u) = \\cos u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img388.svg\" alt=\"$f(u) = \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; é derivável em todo &lt;!-- MATH\n                     $u \\in\n                    \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;; e, além disso,\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} \\cos u = - {\\mathrm {sen}}u.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P202\" title=\"1P202\"&gt;\n                    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img396.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cos u = - {\\mathrm {sen}}u. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"1P203\" title=\"1P203\"&gt;\n                    Conhecendo agora as derivadas de seno e cosseno, definidas em todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, vamos utilizar estas para determinar as\n                    derivadas das outras quatro funções trigonométricas circulares, já que são escritas em termos de seno e cosseno.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"1P204\" title=\"1P204\"&gt;\n                    Consideremos agora a função &lt;a name=\"954\"&gt;&lt;/a&gt; &lt;!-- MATH\n                     $f(u) = {\\mathrm  {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.52ex; vertical-align: -0.85ex; \" src=\"img/img223.svg\" alt=\"$f(u) = {\\mathrm {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida em &lt;!-- MATH\n                     $\\mathbb{R}-\n                    \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como &lt;!-- MATH\n                     ${\\mathrm {sen}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; são diferenciáveis em todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, a derivada do\n                    quociente &lt;!-- MATH\n                     ${\\mathrm  {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.47ex; vertical-align: -0.85ex; \" src=\"img/img114.svg\" alt=\"${\\mathrm {tg}}u = \\frac{{\\mathrm {sen}}u}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt; existe em todos os pontos onde &lt;!-- MATH\n                     $\\cos u \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img397.svg\" alt=\"$\\cos u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"1P205\" title=\"1P205\"&gt;\n                    Dessa forma, para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}- \\{ \\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img188.svg\" alt=\"$u \\in \\mathbb{R}-\\{\\frac{\\pi}{2}+k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P206\" title=\"1P206\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img398.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {tg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.71ex; vertical-align: -1.57ex; \" src=\"img/img399.svg\" alt=\"$\\displaystyle = \\frac{d}{du} \\left( \\tfrac{{\\mathrm {sen}}u}{\\cos u} \\right) = \\frac{({\\mathrm {sen}}u)' \\cos u - {\\mathrm {sen}}u (\\cos u)'}{\\cos^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.67ex; vertical-align: -1.57ex; \" src=\"img/img400.svg\" alt=\"$\\displaystyle = \\frac{\\cos u \\cos u - {\\mathrm {sen}}u (-{\\mathrm {sen}}u)}{\\cos^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.57ex; \" src=\"img/img401.svg\" alt=\"$\\displaystyle = \\frac{\\cos^{2} u + {\\mathrm {sen}}^{2} u}{\\cos^{2} u} = \\frac{1}{\\cos^{2} u} = \\sec^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P207\" title=\"1P207\"&gt;\n                    Portanto\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} {\\mathrm  {tg}}u = \\sec^{2} u,\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P208\" title=\"1P208\"&gt;\n                    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img402.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {tg}}u = \\sec^{2} u, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    &lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P209\" title=\"1P209\"&gt;\n                    para &lt;!-- MATH\n                     $u \\in \\mathbb{R}- \\{ \\tfrac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img403.svg\" alt=\"$u \\in \\mathbb{R}- \\{ \\tfrac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"1P210\" title=\"1P210\"&gt;\n                    Analogamente, para a função &lt;a name=\"986\"&gt;&lt;/a&gt; &lt;!-- MATH\n                     $f(u) = {\\mathrm {ctg}}u = \\frac{\\cos u}{{\\mathrm {sen}}u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.52ex; vertical-align: -0.85ex; \" src=\"img/img236.svg\" alt=\"$f(u) = {\\mathrm {ctg}}u = \\frac{\\cos u}{{\\mathrm {sen}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos para todo &lt;!-- MATH\n                     $u \\in\n                    \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que &lt;!-- MATH\n                     ${\\mathrm {sen}}u \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img237.svg\" alt=\"${\\mathrm {sen}}\n                    u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P211\" title=\"1P211\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img404.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {ctg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.71ex; vertical-align: -1.57ex; \" src=\"img/img405.svg\" alt=\"$\\displaystyle = \\frac{d}{du} \\left( \\tfrac{\\cos u}{{\\mathrm {sen}}u} \\right) = ...\n                    ...(\\cos u)' {\\mathrm {sen}}u - \\cos u ({\\mathrm {sen}}u)'}{{\\mathrm {sen}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.32ex; vertical-align: -1.57ex; \" src=\"img/img406.svg\" alt=\"$\\displaystyle = \\frac{-{\\mathrm {sen}}u {\\mathrm {sen}}u - \\cos u \\cos u}{{\\mathrm {sen}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.57ex; \" src=\"img/img407.svg\" alt=\"$\\displaystyle = \\frac{-({\\mathrm {sen}}^{2} u + \\cos^{2} u)}{{\\mathrm {sen}}^{2} u} = \\frac{-1}{{\\mathrm {sen}}^{2} u} = -\\csc^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P212\" title=\"1P212\"&gt;\n                    Então,\n                    &lt;/p&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{d}{du} {\\mathrm {ctg}}u = \\csc^{2} u,\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"1P213\" title=\"1P213\"&gt;\n                    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img408.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {ctg}}u = \\csc^{2} u, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P214\" title=\"1P214\"&gt;\n                    para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img191.svg\" alt=\"$u \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"1P215\" title=\"1P215\"&gt;\n                    Considerando agora a função &lt;a name=\"1010\"&gt;&lt;/a&gt; &lt;!-- MATH\n                     $f(u) = \\sec u = \\frac{1}{\\cos u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img409.svg\" alt=\"$f(u) = \\sec u = \\frac{1}{\\cos u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal\n                    que &lt;!-- MATH\n                     $\\cos u \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img397.svg\" alt=\"$\\cos u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P216\" title=\"1P216\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img410.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.71ex; vertical-align: -1.57ex; \" src=\"img/img411.svg\" alt=\"$\\displaystyle = \\frac{d}{du} \\left( \\tfrac{1}{\\cos u} \\right) = \\frac{(1)' \\cos u - (-{\\mathrm {sen}}u)}{\\cos^{2}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img412.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}u}{\\cos^{2}u} = \\frac{{\\mathrm {sen}}u}{\\cos u} \\cdot \\frac{1}{\\cos u} = {\\mathrm {tg}}u \\sec u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"1P217\" title=\"1P217\"&gt;\n                    Da mesma forma, para a função &lt;a name=\"1029\"&gt;&lt;/a&gt; &lt;!-- MATH\n                     $f(u) = \\csc u = \\frac{1}{{\\mathrm {sen}}u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img413.svg\" alt=\"$f(u) = \\csc u = \\frac{1}{{\\mathrm {sen}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos que\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"1P218\" title=\"1P218\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img414.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.71ex; vertical-align: -1.57ex; \" src=\"img/img415.svg\" alt=\"$\\displaystyle = \\frac{d}{du} \\left( \\tfrac{1}{{\\mathrm {sen}}u} \\right) = \\frac{(1)' {\\mathrm {sen}}u - \\cos u}{{\\mathrm {sen}}^{2}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img416.svg\" alt=\"$\\displaystyle = -\\frac{\\cos u}{{\\mathrm {sen}}^{2}u} = -\\frac{\\cos u}{{\\mathrm {sen}}u} \\cdot \\frac{1}{{\\mathrm {sen}}u} = -{\\mathrm {ctg}}u \\csc u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" style=\"text-indent: 0px !important;\" id=\"1P219\" title=\"1P219\"&gt;\n                    para todo &lt;!-- MATH\n                     $u \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que &lt;!-- MATH\n                     ${\\mathrm {sen}}u \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img237.svg\" alt=\"${\\mathrm {sen}}\n                    u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"1P220\" title=\"1P220\"&gt;\n                    Vamos resumir as fórmulas de derivação para as funções trigonométricas circulares em uma tabela.\n                    &lt;/p&gt;\n                    \n                    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1062\"&gt;&lt;/a&gt;\n                    &lt;table id=\"1T2\" title=\"1T2\"&gt;\n                    &lt;caption&gt;&lt;strong&gt;Tabela 1.2:&lt;/strong&gt;\n                    Derivadas das funções trigonométricas circulares.&lt;/caption&gt;\n                    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                    &lt;div class=\"CENTER\"&gt;\n                    &lt;table class=\"PAD  BORDER\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;derivada&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     ${\\mathrm {sen}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img228.svg\" alt=\"${\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $\\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img227.svg\" alt=\"$\\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $\\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $-{\\mathrm {sen}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img417.svg\" alt=\"$-{\\mathrm {sen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     ${\\mathrm  {tg}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img281.svg\" alt=\"${\\mathrm {tg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                     $\\mathbb{R}- \\{ \\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $\\sec^{2} u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img418.svg\" alt=\"$\\sec^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     ${\\mathrm {ctg}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img115.svg\" alt=\"${\\mathrm {ctg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $-\\csc^{2} u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img419.svg\" alt=\"$-\\csc^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img121.svg\" alt=\"$\\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                     $\\mathbb{R}- \\{ \\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img225.svg\" alt=\"$\\mathbb{R}- \\{\\frac{\\pi}{2} + k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     ${\\mathrm  {tg}}u \\sec u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img420.svg\" alt=\"${\\mathrm {tg}}u \\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img123.svg\" alt=\"$\\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img238.svg\" alt=\"$\\mathbb{R}- \\{ k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                     $-{\\mathrm {ctg}}u \\csc u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.90ex; vertical-align: -0.58ex; \" src=\"img/img421.svg\" alt=\"$-{\\mathrm {ctg}}u \\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;\n                    &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;\n                    &lt;/div&gt;\n                    &lt;br&gt;",
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      "Capítulo 1​: Funções trigonométricas circulares​"
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.5 As funções trigonométricas circulares inversas",
    "text": "1.5 As funções trigonométricas circulares inversas\n\n    &lt;p class=\" unidade\" id=\"1P221\" title=\"1P221\"&gt;\n    Antes de começarmos, lembremos que nenhuma das funções trigonométricas circulares é bijetora dos seus domínios de\n    definição em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Sabemos também que somente as funções bijetoras possuem função inversa. O que precisamos então é\n    restringir o domínio e/ou o contradomínio de tais funções, conforme o caso exigir, a fim de torná-las bijetoras e só\n    então poderemos definir as funções inversas.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P222\" title=\"1P222\"&gt;\n    Comecemos então pela função &lt;a name=\"1066\"&gt;&lt;/a&gt; seno que não é injetora e nem sobrejetora de &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em\n    &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Resolvemos o problema da sobrejetividade restringindo o contradomínio, tornando-o igual à imagem\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;. O problema da injetividade é resolvido restringindo o domínio. Consideramos então o domínio como sendo\n    &lt;!-- MATH\n     $[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img422.svg\" alt=\"$[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;. A função seno é bijetora de &lt;!-- MATH\n     $[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img422.svg\" alt=\"$[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Podemos\n    definir então a função inversa do seno, que a cada número real &lt;!-- MATH\n     $u \\in [-1,1]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; associa o (único) número &lt;!-- MATH\n     $w \\in\n    [-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img424.svg\" alt=\"$w \\in\n    [-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;, satisfazendo a relação &lt;!-- MATH\n     $u = {\\mathrm {sen}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img425.svg\" alt=\"$u = {\\mathrm {sen}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P223\" title=\"1P223\"&gt;\n    Representamos a função inversa do seno por por &lt;!-- MATH\n     $w = f(u) = {\\mathrm {sen}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img426.svg\" alt=\"$w = f(u) = {\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou &lt;!-- MATH\n     $w = {\\mathrm{arcsen}}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img427.svg\" alt=\"$w = {\\mathrm{arcsen}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Neste texto, vamos utilizar\n    a primeira notação e muito cuidado para não confundir as expressões &lt;!-- MATH\n     ${\\mathrm {sen}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img428.svg\" alt=\"${\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n     $({\\mathrm {sen}}u)^{-1}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img429.svg\" alt=\"$({\\mathrm {sen}}u)^{-1}$\" loading=\"lazy\"&gt;&lt;/span&gt;. A segunda\n    expressão é o inverso multiplicativo do seno, ou seja &lt;!-- MATH\n     $({\\mathrm {sen}}u)^{-1} = \\frac{1}{{\\mathrm {sen}}u} = \\csc u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img430.svg\" alt=\"$({\\mathrm {sen}}u)^{-1} = \\frac{1}{{\\mathrm {sen}}u} = \\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim, temos\n    definida a função\n    &lt;/p&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P224\" title=\"1P224\"&gt;\n    &lt;!-- MATH\n     \\begin{eqnarray*}\n    f : [-1,1] & \\to & [-\\tfrac{\\pi}{2}, \\tfrac{\\pi}{2}] \\\\\n    u & \\mapsto & w = f(u) = {\\mathrm {sen}}^{-1}u\n    \\end{eqnarray*}\n     --&gt;\n    &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n    &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img431.svg\" alt=\"$\\displaystyle f : [-1,1]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img432.svg\" alt=\"$\\displaystyle [-\\tfrac{\\pi}{2}, \\tfrac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img433.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    &lt;p class=\" unidade\" id=\"1P225\" style=\"text-indent: 0px !important;\" title=\"1P225\"&gt;\n    que deve satisfazer a relação &lt;!-- MATH\n     $u = {\\mathrm {sen}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img425.svg\" alt=\"$u = {\\mathrm {sen}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta igualdade, vemos que quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela\n    esquerda), devemos ter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximando de &lt;!-- MATH\n     $\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; e quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; tender a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela direita), devemos\n    ter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; tendendo a &lt;!-- MATH\n     $-\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img435.svg\" alt=\"$-\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto é,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -1^{+}} {\\mathrm {sen}}^{-1}u = -\\frac{\\pi}{2} \\qquad \\text{e} \\qquad \\lim_{u \\to 1^{-}} {\\mathrm {sen}}^{-1}u = \\frac{\\pi}{2}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P226\" title=\"1P226\"&gt;\n    &lt;img style=\"height: 4.41ex; vertical-align: -1.93ex; \" src=\"img/img436.svg\" alt=\"$\\displaystyle \\lim_{u \\to -1^{+}} {\\mathrm {sen}}^{-1}u = -\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.29ex; vertical-align: -1.81ex; \" src=\"img/img437.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 1^{-}} {\\mathrm {sen}}^{-1}u = \\frac{\\pi}{2}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P227\" title=\"1P227\"&gt;\n    Deve ser uma função crescente no intervalo de definição &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;, já que da relação &lt;!-- MATH\n     $u = {\\mathrm {sen}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img425.svg\" alt=\"$u = {\\mathrm {sen}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; vemos que conforme &lt;!-- MATH\n     $u\n    \\in [-1,1]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce, o ângulo radiano &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve também crescer em &lt;!-- MATH\n     $-[\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img438.svg\" alt=\"$-[\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como é de se esperar\n    de uma função inversa, são válidas as seguintes relações,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P228\" title=\"1P228\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img439.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}({\\mathrm {sen}}^{-1} u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img441.svg\" alt=\"$\\displaystyle \\quad u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img442.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}^{-1}({\\mathrm {sen}}u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img443.svg\" alt=\"$\\displaystyle \\quad u \\in [-\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}].$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P229\" title=\"1P229\"&gt;\n    O gráfico desta função, definida apenas no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;, é dado por\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1123\"&gt;&lt;/a&gt;\n    &lt;table id=\"1I18\" title=\"1I18\"&gt;\n    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.18:&lt;/strong&gt;\n    Gráfico da função seno inverso.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;img src=\"img/./farcsin.png\" alt=\"Image farcsin\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P230\" title=\"1P230\"&gt;\n    Agora tomemos a função &lt;a name=\"1125\"&gt;&lt;/a&gt; cosseno. Sabemos que a função cosseno, também não é injetora e\n    nem sobrejetora de &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Colocando o contradomínio como sendo a imagem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; a tornamos sobrejetora.\n    Colocando o domínio como sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img444.svg\" alt=\"$[0,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt; a tornamos injetora de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img444.svg\" alt=\"$[0,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Definimos assim, a função\n    inversa do cosseno, denotada por &lt;!-- MATH\n     $f(u) = \\cos^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img445.svg\" alt=\"$f(u) = \\cos^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; (ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img446.svg\" alt=\"$\\arccos u$\" loading=\"lazy\"&gt;&lt;/span&gt;), por\n    &lt;/p&gt;\n     \n    &lt;div class=\"mathdisplay unidade\" id=\"1P231\" title=\"1P231\"&gt;\n    &lt;!-- MATH\n     \\begin{eqnarray*}\n    f : [-1,1] & \\to & [0,\\pi] \\\\\n    u & \\mapsto & w = f(u) = \\cos^{-1}u\n    \\end{eqnarray*}\n     --&gt;\n    &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n    &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img431.svg\" alt=\"$\\displaystyle f : [-1,1]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img447.svg\" alt=\"$\\displaystyle [0,\\pi]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img448.svg\" alt=\"$\\displaystyle w = f(u) = \\cos^{-1}u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P232\" style=\"text-indent: 0px !important;\" title=\"1P232\"&gt;\n    que deve satisfazer &lt;!-- MATH\n     $u = \\cos w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img449.svg\" alt=\"$u = \\cos w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Usando esta última igualdade, vemos que quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; estiver suficientemente próximo de\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela direita) então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; estará próximo de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt; e quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; estiver suficientemente próximo de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    (somente pela esquerda), então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; estará próximo de &lt;span class=\"MATH\"&gt;0&lt;/span&gt;. Valem portanto os limites,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -1^{+}} \\cos^{-1}u = \\pi \\qquad \\text{e} \\qquad \\lim_{u \\to 1^{-}} \\cos^{-1}u = 0.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P233\" title=\"1P233\"&gt;\n    &lt;img style=\"height: 3.96ex; vertical-align: -1.93ex; \" src=\"img/img450.svg\" alt=\"$\\displaystyle \\lim_{u \\to -1^{+}} \\cos^{-1}u = \\pi$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.84ex; vertical-align: -1.81ex; \" src=\"img/img451.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 1^{-}} \\cos^{-1}u = 0. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P234\" title=\"1P234\"&gt;\n    O gráfico desta função é dado por,\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1137\"&gt;&lt;/a&gt;\n    &lt;table id=\"1I19\" title=\"1I19\"&gt;\n    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.19:&lt;/strong&gt;\n    Gráfico da função cosseno inverso.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;img src=\"img/./farccos.png\" alt=\"Image farccos\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P235\" title=\"1P235\"&gt;\n    É uma função decrescente no intervalo de definição. Analogamente, as relações inversas são,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P236\" title=\"1P236\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img452.svg\" alt=\"$\\displaystyle \\cos(\\cos^{-1} u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img441.svg\" alt=\"$\\displaystyle \\quad u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img453.svg\" alt=\"$\\displaystyle \\cos^{-1}(\\cos u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img454.svg\" alt=\"$\\displaystyle \\quad u \\in [0,\\pi].$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P237\" title=\"1P237\"&gt;\n    Note que existe uma relação entre os gráficos das funções seno e cosseno inversas. Se tomarmos o gráfico da função seno\n    inverso e aplicarmos uma reflexão em torno do eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; e um deslocamento de &lt;!-- MATH\n     $\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; unidades para cima,\n    teremos o gráfico da função cosseno inverso. Esta relação é descrita pela igualdade &lt;!-- MATH\n     $\\cos^{-1} u = \\frac{\\pi}{2} -\n    {\\mathrm {sen}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img455.svg\" alt=\"$\\cos^{-1} u = \\frac{\\pi}{2} -\n    {\\mathrm {sen}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P238\" title=\"1P238\"&gt;\n    É fácil provar esta última igualdade usando que &lt;!-- MATH\n     $\\cos w = - {\\mathrm {sen}}(w - \\frac{\\pi}{2}) = {\\mathrm {sen}}(\\frac{\\pi}{2} - w)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img456.svg\" alt=\"$\\cos w = - {\\mathrm {sen}}(w - \\frac{\\pi}{2}) = {\\mathrm {sen}}(\\frac{\\pi}{2} - w)$\" loading=\"lazy\"&gt;&lt;/span&gt; é válido\n    para todo &lt;!-- MATH\n     $w \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img457.svg\" alt=\"$w \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Vejamos os detalhes. Primeiro observemos que a função &lt;!-- MATH\n     $w = {\\mathrm {sen}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img458.svg\" alt=\"$w = {\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função ímpar, pois\n    se &lt;!-- MATH\n     $w = {\\mathrm {sen}}^{-1}(-u)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img459.svg\" alt=\"$w = {\\mathrm {sen}}^{-1}(-u)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então &lt;!-- MATH\n     ${\\mathrm {sen}}w = -u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img460.svg\" alt=\"${\\mathrm {sen}}w = -u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, &lt;!-- MATH\n     $u = -{\\mathrm {sen}}w = {\\mathrm {sen}}(-w)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img461.svg\" alt=\"$u = -{\\mathrm {sen}}w = {\\mathrm {sen}}(-w)$\" loading=\"lazy\"&gt;&lt;/span&gt;, o que acarreta &lt;!-- MATH\n     $-w = {\\mathrm {sen}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img462.svg\" alt=\"$-w = {\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n    segue que &lt;!-- MATH\n     ${\\mathrm {sen}}^{-1}(-u) = w = -{\\mathrm {sen}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img463.svg\" alt=\"${\\mathrm {sen}}^{-1}(-u) = w = -{\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Agora vamos à igualdade de interesse. Se &lt;!-- MATH\n     $u = \\cos w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img449.svg\" alt=\"$u = \\cos w$\" loading=\"lazy\"&gt;&lt;/span&gt;, então temos que &lt;!-- MATH\n     $-u\n    = -\\cos w = {\\mathrm {sen}}(w - \\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img464.svg\" alt=\"$-u\n    = -\\cos w = {\\mathrm {sen}}(w - \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;, donde &lt;!-- MATH\n     $w - \\frac{\\pi}{2} = {\\mathrm {sen}}^{-1}(-u)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img465.svg\" alt=\"$w - \\frac{\\pi}{2} = {\\mathrm {sen}}^{-1}(-u)$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, &lt;!-- MATH\n     $w = \\frac{\\pi}{2} +\n    {\\mathrm {sen}}^{-1}(-u) = \\frac{\\pi}{2} - {\\mathrm {sen}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img466.svg\" alt=\"$w = \\frac{\\pi}{2} +\n    {\\mathrm {sen}}^{-1}(-u) = \\frac{\\pi}{2} - {\\mathrm {sen}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como também &lt;!-- MATH\n     $w = \\cos^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img467.svg\" alt=\"$w = \\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt; então &lt;!-- MATH\n     $\\cos^{-1} u = \\frac{\\pi}{2} -\n    {\\mathrm {sen}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img455.svg\" alt=\"$\\cos^{-1} u = \\frac{\\pi}{2} -\n    {\\mathrm {sen}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P239\" title=\"1P239\"&gt;\n    Consideremos agora a função &lt;a name=\"1176\"&gt;&lt;/a&gt; tangente, que é sobrejetora, porém não é injetora de &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para resolver o problema da injetividade precisamos restringir somente o domínio desta função. Considerando\n    então o domínio como sendo o intervalo &lt;!-- MATH\n     $(-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img468.svg\" alt=\"$(-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt; temos a bijetividade da função tangente, de\n    &lt;!-- MATH\n     $(-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img468.svg\" alt=\"$(-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Definimos então a função inversa da tangente, denotada por &lt;!-- MATH\n     $f(u) =\n    {\\mathrm  {tg}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img469.svg\" alt=\"$f(u) =\n    {\\mathrm {tg}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; (ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img470.svg\" alt=\"$\\arctan u$\" loading=\"lazy\"&gt;&lt;/span&gt;), como\n    &lt;/p&gt;\n     \n    &lt;div class=\"mathdisplay unidade\" id=\"1P240\" title=\"1P240\"&gt;\n    &lt;!-- MATH\n     \\begin{eqnarray*}\n    f : \\mathbb{R}& \\to & (-\\tfrac{\\pi}{2}, \\tfrac{\\pi}{2}) \\\\\n    u & \\mapsto & w = f(u) = {\\mathrm  {tg}}^{-1}u\n    \\end{eqnarray*}\n     --&gt;\n    &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n    &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img210.svg\" alt=\"$\\displaystyle f : \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img471.svg\" alt=\"$\\displaystyle (-\\tfrac{\\pi}{2}, \\tfrac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img472.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm {tg}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n   \n    &lt;p class=\" unidade\" id=\"1P241\" style=\"text-indent: 0px !important;\" title=\"1P241\"&gt;\n    de tal forma que vale a relação &lt;!-- MATH\n     ${\\mathrm  {tg}}w = u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img473.svg\" alt=\"${\\mathrm {tg}}w = u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta relação mostra que quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce indefinidamente então devemos\n    ter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximando de &lt;!-- MATH\n     $\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; e quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce indefinidamente, com valores negativos, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve\n    estar se aproximando de &lt;!-- MATH\n     $-\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img435.svg\" alt=\"$-\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto se resume nos limites\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -\\infty} {\\mathrm  {tg}}^{-1} u = -\\frac{\\pi}{2} \\qquad \\text{e} \\qquad \\lim_{u \\to \\infty} {\\mathrm  {tg}}^{-1} u = \\frac{\\pi}{2}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P242\" title=\"1P242\"&gt;\n    &lt;img style=\"height: 4.21ex; vertical-align: -1.73ex; \" src=\"img/img474.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} {\\mathrm {tg}}^{-1} u = -\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.09ex; vertical-align: -1.60ex; \" src=\"img/img475.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to \\infty} {\\mathrm {tg}}^{-1} u = \\frac{\\pi}{2}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P243\" title=\"1P243\"&gt;\n    O gráfico desta função é dado por,\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1208\"&gt;&lt;/a&gt;\n    &lt;table id=\"1I20\" title=\"1I20\"&gt;\n    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.20:&lt;/strong&gt;\n    Gráfico da função tangente inversa.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;img src=\"img/./farctan.png\" alt=\"Image farctan\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P244\" title=\"1P244\"&gt;\n    É uma função monótona crescente e ímpar. As retas &lt;!-- MATH\n     $w = \\pm \\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img476.svg\" alt=\"$w = \\pm \\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; são assíntotas horizontais desta função. Esta\n    função desempenha um papel importante na matemática. Ela associa bijetivamente toda a reta real com um intervalo\n    limitado. Valem as relações inversas,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P245\" title=\"1P245\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img477.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}({\\mathrm {tg}}^{-1} u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img478.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img479.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}({\\mathrm {tg}}u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img480.svg\" alt=\"$\\displaystyle \\quad u \\in (-\\tfrac{\\pi}{2}, \\tfrac{\\pi}{2}).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P246\" title=\"1P246\"&gt;\n    Agora a função &lt;a name=\"1222\"&gt;&lt;/a&gt; cotangente. Vimos que a cotangente não é uma função injetora, mas é\n    sobrejetora de &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. A restrição do domínio para o intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img481.svg\" alt=\"$(0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; faz da função cotangente, uma função\n    bijetora de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img481.svg\" alt=\"$(0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim podemos definir a função cotangente inversa, denotada por &lt;!-- MATH\n     $f(u) = {\\mathrm {ctg}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img482.svg\" alt=\"$f(u) = {\\mathrm {ctg}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n    dada por\n    &lt;/p&gt;\n \n    &lt;div class=\"mathdisplay unidade\" id=\"1P247\" title=\"1P247\"&gt;\n    &lt;!-- MATH\n     \\begin{eqnarray*}\n    f : \\mathbb{R}& \\to & (0,\\pi) \\\\\n    u & \\mapsto & w = f(u) = {\\mathrm {ctg}}^{-1} u\n    \\end{eqnarray*}\n     --&gt;\n    &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n    &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img210.svg\" alt=\"$\\displaystyle f : \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img483.svg\" alt=\"$\\displaystyle (0,\\pi)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img484.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n \n    &lt;p class=\" unidade\" id=\"1P248\" style=\"text-indent: 0px !important;\" title=\"1P248\"&gt;\n    desde que valha a relação &lt;!-- MATH\n     $u = {\\mathrm {ctg}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img485.svg\" alt=\"$u = {\\mathrm {ctg}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta relação mostra que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; tende ao infinito, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar indo\n    para &lt;span class=\"MATH\"&gt;0&lt;/span&gt;; e se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; vai para o infinito negativo, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar indo para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos assim,\n     &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -\\infty} {\\mathrm {ctg}}^{-1} u = \\pi \\qquad \\text{e} \\qquad \\lim_{u \\to \\infty} {\\mathrm {ctg}}^{-1} u = 0.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P249\" title=\"1P249\"&gt;\n    &lt;img style=\"height: 3.76ex; vertical-align: -1.73ex; \" src=\"img/img486.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} {\\mathrm {ctg}}^{-1} u = \\pi$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.63ex; vertical-align: -1.60ex; \" src=\"img/img487.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to \\infty} {\\mathrm {ctg}}^{-1} u = 0. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P250\" title=\"1P250\"&gt;\n    O gráfico da função cotangente inversa é\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1234\"&gt;&lt;/a&gt;\n    &lt;table id=\"1I21\" title=\"1I21\"&gt;\n    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.21:&lt;/strong&gt;\n    Gráfico da função cotangente inversa.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;img src=\"img/./farcctg.png\" alt=\"Image farcctg\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P251\" title=\"1P251\"&gt;\n    Vemos que é uma função estritamente decrescente. Além disso, são válidas as relações inversas,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P252\" title=\"1P252\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img488.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}({\\mathrm {ctg}}^{-1} u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img478.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img489.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}^{-1}({\\mathrm {ctg}}u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img490.svg\" alt=\"$\\displaystyle \\quad u \\in (0,\\pi).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P253\" title=\"1P253\"&gt;\n    Assim como no caso do cosseno inverso, existe uma relação entre os gráficos das funções tangente e cotangente inversas.\n    Esta relação é semelhante àquela envolvendo seno e cosseno inversos. É &lt;!-- MATH\n     ${\\mathrm {ctg}}^{-1} u = \\frac{\\pi}{2} - {\\mathrm  {tg}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img491.svg\" alt=\"${\\mathrm {ctg}}^{-1} u = \\frac{\\pi}{2} - {\\mathrm {tg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    Também é fácil provar esta relação usando a igualdade &lt;!-- MATH\n     ${\\mathrm {ctg}}w = - {\\mathrm  {tg}}(w - \\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img492.svg\" alt=\"${\\mathrm {ctg}}w = - {\\mathrm {tg}}(w - \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;, que é válida para todo &lt;!-- MATH\n     $w\n    \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img493.svg\" alt=\"$w\n    \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta vez deixamos os detalhes por conta do leitor.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P254\" title=\"1P254\"&gt;\n    Tomando a função &lt;a name=\"1248\"&gt;&lt;/a&gt; secante, lembremos que ela não é injetora e nem sobrejetora de &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em\n    &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. A imagem é o conjunto &lt;!-- MATH\n     $(-\\infty,-1] \\cup [1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; e então restringimos o contradomínio no conjunto imagem e\n    tornamos a secante sobrejetora. Para a injetividade, escolhemos o conjunto &lt;!-- MATH\n     $[0,\\pi] - \\{\\frac{\\pi}{2}\\} = [0,\n    \\frac{\\pi}{2}) \\cup (\\frac{\\pi}{2},\\pi]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img494.svg\" alt=\"$[0,\\pi] - \\{\\frac{\\pi}{2}\\} = [0,\n    \\frac{\\pi}{2}) \\cup (\\frac{\\pi}{2},\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Nestes termos a função secante, é bijetora de &lt;!-- MATH\n     $[0, \\frac{\\pi}{2}) \\cup\n    (\\frac{\\pi}{2},\\pi]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img495.svg\" alt=\"$[0, \\frac{\\pi}{2}) \\cup\n    (\\frac{\\pi}{2},\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n     $(-\\infty,-1] \\cup [1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Definimos então a função secante inversa, denotada por &lt;!-- MATH\n     $f(u) =\n    \\sec^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img496.svg\" alt=\"$f(u) =\n    \\sec^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, como\n    &lt;/p&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P255\" title=\"1P255\"&gt;\n    &lt;!-- MATH\n     \\begin{eqnarray*}\n    f : (-\\infty,-1] \\cup [1,\\infty) & \\to & [0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi] \\\\\n    u & \\mapsto & w = f(u) = \\sec^{-1} u\n    \\end{eqnarray*}\n     --&gt;\n    &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n    &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img497.svg\" alt=\"$\\displaystyle f : (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img498.svg\" alt=\"$\\displaystyle [0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img499.svg\" alt=\"$\\displaystyle w = f(u) = \\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P256\" style=\"text-indent: 0px !important;\" title=\"1P256\"&gt;\n    desde que valha a relação &lt;!-- MATH\n     $\\sec w = u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img500.svg\" alt=\"$\\sec w = u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta relação vemos pelo gráfico da função secante que quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; vai ao\n    infinito positivamente ou negativamente então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar se aproximando de &lt;!-- MATH\n     $\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto é,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -\\infty} \\sec^{-1} u = \\lim_{u \\to \\infty} \\sec^{-1} u = \\frac{\\pi}{2}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P257\" title=\"1P257\"&gt;\n    &lt;img style=\"height: 4.21ex; vertical-align: -1.73ex; \" src=\"img/img501.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} \\sec^{-1} u = \\lim_{u \\to \\infty} \\sec^{-1} u = \\frac{\\pi}{2}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P258\" title=\"1P258\"&gt;\n    Também quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela esquerda), devemos ter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximando de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\" loading=\"lazy\"&gt;&lt;/span&gt; e quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    tende a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela direita) &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar indo para &lt;span class=\"MATH\"&gt;0&lt;/span&gt;. Temos então os limites\n     &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -1^{-}} \\sec^{-1} u = \\pi \\qquad \\text{e} \\qquad \\lim_{u \\to 1^{+}} \\sec^{-1} u = 0.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P259\" title=\"1P259\"&gt;\n    &lt;img style=\"height: 3.96ex; vertical-align: -1.93ex; \" src=\"img/img502.svg\" alt=\"$\\displaystyle \\lim_{u \\to -1^{-}} \\sec^{-1} u = \\pi$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.84ex; vertical-align: -1.81ex; \" src=\"img/img503.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 1^{+}} \\sec^{-1} u = 0. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P260\" title=\"1P260\"&gt;\n    O gráfico desta função é da forma,\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1282\"&gt;&lt;/a&gt;\n    &lt;table id=\"1I22\" title=\"1I22\"&gt;\n    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.22:&lt;/strong&gt;\n    Gráfico da função secante inversa.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;img src=\"img/./farcsec.png\" alt=\"Image farcsec\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P261\" title=\"1P261\"&gt;\n    Para esta função, valem as relações inversas,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P262\" title=\"1P262\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img504.svg\" alt=\"$\\displaystyle \\sec(\\sec^{-1} u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img505.svg\" alt=\"$\\displaystyle \\quad u \\in (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img506.svg\" alt=\"$\\displaystyle \\sec^{-1}(\\sec u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img507.svg\" alt=\"$\\displaystyle \\quad u \\in [0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi].$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P263\" title=\"1P263\"&gt;\n    Finalmente, a função &lt;a name=\"1294\"&gt;&lt;/a&gt; cossecante não é injetora nem sobrejetora de &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    Restringimos então o contradomínio pela sua imagem, que é o conjunto &lt;!-- MATH\n     $(-\\infty,-1] \\cup [1, \\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para acertar a\n    injetividade, escolhemos a restrição do domínio ao conjunto &lt;!-- MATH\n     $[-\\frac{\\pi}{2}, \\frac{\\pi}{2}] - \\{0\\} =\n    [-\\frac{\\pi}{2},0) \\cup (0,\\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img508.svg\" alt=\"$[-\\frac{\\pi}{2}, \\frac{\\pi}{2}] - \\{0\\} =\n    [-\\frac{\\pi}{2},0) \\cup (0,\\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim a função cossecante se tornará bijetiva. Então definimos a função\n    cossecante inversa, denotada por &lt;!-- MATH\n     $f(u) = \\csc^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img509.svg\" alt=\"$f(u) = \\csc^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e dada por\n     &lt;/p&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P264\" title=\"1P264\"&gt;\n    &lt;!-- MATH\n     \\begin{eqnarray*}\n    f: (-\\infty,-1] \\cup [1, \\infty) & \\to & [-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2}] \\\\\n    u & \\mapsto & w = f(u) =\\csc^{-1} u\n    \\end{eqnarray*}\n     --&gt;\n    &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n    &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img497.svg\" alt=\"$\\displaystyle f : (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img510.svg\" alt=\"$\\displaystyle [-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img511.svg\" alt=\"$\\displaystyle w = f(u) =\\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n    &nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n  \n    &lt;p class=\" unidade\" id=\"1P265\" style=\"text-indent: 0px !important;\" title=\"1P265\"&gt;\n    de tal forma que &lt;!-- MATH\n     $\\csc w = u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img512.svg\" alt=\"$\\csc w = u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta relação, observando o gráfico da função cossecante, temos que quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce\n    indefinidamente (positivamente ou negativamente), os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; devem estar se aproximando de &lt;span class=\"MATH\"&gt;0&lt;/span&gt;. Por isto temos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -\\infty} \\csc^{-1} u = \\lim_{u \\to \\infty} \\csc^{-1} u = 0.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P266\" title=\"1P266\"&gt;\n    &lt;img style=\"height: 3.76ex; vertical-align: -1.73ex; \" src=\"img/img513.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} \\csc^{-1} u = \\lim_{u \\to \\infty} \\csc^{-1} u = 0. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P267\" title=\"1P267\"&gt;\n    Também se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img249.svg\" alt=\"$-1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela esquerda), devemos ter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximando de &lt;!-- MATH\n     $-\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img435.svg\" alt=\"$-\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Da mesma\n    forma, se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; (somente pela direita) então devemos ter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximando de &lt;!-- MATH\n     $\\frac{\\pi}{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\lim_{u \\to -1^{-}} \\csc^{-1} u = -\\frac{\\pi}{2} \\qquad \\text{e} \\qquad \\lim_{u \\to 1^{+}} \\csc^{-1} u = \\frac{\\pi}{2}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P268\" title=\"1P268\"&gt;\n    &lt;img style=\"height: 4.41ex; vertical-align: -1.93ex; \" src=\"img/img514.svg\" alt=\"$\\displaystyle \\lim_{u \\to -1^{-}} \\csc^{-1} u = -\\frac{\\pi}{2}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.29ex; vertical-align: -1.81ex; \" src=\"img/img515.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 1^{+}} \\csc^{-1} u = \\frac{\\pi}{2}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P269\" title=\"1P269\"&gt;\n    O gráfico desta função é da forma,\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1330\"&gt;&lt;/a&gt;\n    &lt;table id=\"1I23\" title=\"1I23\"&gt;\n    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 1.23:&lt;/strong&gt;\n    Gráfico da função cossecante inversa.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;img src=\"img/./farccsc.png\" alt=\"Image farccsc\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P270\" title=\"1P270\"&gt;\n    Para esta função, valem as relações inversas são,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P271\" title=\"1P271\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img516.svg\" alt=\"$\\displaystyle \\csc(\\csc^{-1} u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img505.svg\" alt=\"$\\displaystyle \\quad u \\in (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img517.svg\" alt=\"$\\displaystyle \\csc^{-1}(\\csc u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img440.svg\" alt=\"$\\displaystyle = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img518.svg\" alt=\"$\\displaystyle \\quad u \\in [-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2}].$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P272\" title=\"1P272\"&gt;\n    Novamente vemos a presença de uma relação entre os gráficos de secante e cossecante inversas. É novamente a igualdade\n    &lt;!-- MATH\n     $\\csc^{-1} u = \\frac{\\pi}{2} - \\sec^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img519.svg\" alt=\"$\\csc^{-1} u = \\frac{\\pi}{2} - \\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Podemos provar esta igualdade usando &lt;!-- MATH\n     $\\csc w = \\sec(\\frac{\\pi}{2}-w)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img520.svg\" alt=\"$\\csc w = \\sec(\\frac{\\pi}{2}-w)$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n    válida para &lt;!-- MATH\n     $w \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img493.svg\" alt=\"$w\n    \\in \\mathbb{R}-\\{k\\pi; \\, k \\in \\mathbb{Z}\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Detalhes novamente por conta do leitor.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P273\" title=\"1P273\"&gt;\n    Resumimos as funções trigonométricas circulares inversas, com seus respectivos domínios e imagens na próxima tabela.\n    \n    &lt;/p&gt;\n    \n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1378\"&gt;&lt;/a&gt;\n    &lt;table id=\"1T3\" title=\"1T3\"&gt;\n    &lt;caption&gt;&lt;strong&gt;Tabela 1.3:&lt;/strong&gt;\n    Domínio e imagem das funções trigonométricas circulares inversas&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;table class=\"PAD  BORDER\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;imagem&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     ${\\mathrm {sen}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img428.svg\" alt=\"${\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $[-\\frac{\\pi}{2},\\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img422.svg\" alt=\"$[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\cos^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img521.svg\" alt=\"$\\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img444.svg\" alt=\"$[0,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     ${\\mathrm  {tg}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img522.svg\" alt=\"${\\mathrm {tg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $(-\\frac{\\pi}{2},\\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img468.svg\" alt=\"$(-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     ${\\mathrm {ctg}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img523.svg\" alt=\"${\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img481.svg\" alt=\"$(0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\sec^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img524.svg\" alt=\"$\\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n     $(-\\infty,-1] \\cup [1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $[0,\\frac{\\pi}{2}) \\cup (\\frac{\\pi}{2},\\pi]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img495.svg\" alt=\"$[0, \\frac{\\pi}{2}) \\cup\n    (\\frac{\\pi}{2},\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\csc^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img525.svg\" alt=\"$\\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n     $(-\\infty,-1] \\cup [1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img251.svg\" alt=\"$(-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $[-\\frac{\\pi}{2},0) \\cup (0,\\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img526.svg\" alt=\"$[-\\frac{\\pi}{2},0) \\cup (0,\\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    &lt;br&gt;",
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      "Capítulo 1​: Funções trigonométricas circulares​"
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.6 Continuidade das funções trigonométricas circulares inversas",
    "text": "1.6 Continuidade das funções trigonométricas circulares inversas\n\n        &lt;p class=\" unidade\" id=\"1P274\" title=\"1P274\"&gt;\n        Antes de obtermos as derivadas das funções trigonométricas circulares inversas, vamos analisar a continuidade destas\n        funções em todos os pontos de definição. Esta continuidade pode ser obtida em virtude da continuidade das funções\n        trigonométricas circulares estabelecida na seção &lt;a href=\"#seccftc\"&gt;1.3&lt;/a&gt;. Mais precisamente se &lt;!-- MATH\n         $I \\subset \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img527.svg\" alt=\"$I \\subset \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; é um intervalo e\n        &lt;!-- MATH\n         $f: I \\to J$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.91ex; vertical-align: -0.36ex; \" src=\"img/img528.svg\" alt=\"$f: I \\to J$\" loading=\"lazy\"&gt;&lt;/span&gt; é contínua em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.91ex; vertical-align: -0.36ex; \" src=\"img/img530.svg\" alt=\"$J$\" loading=\"lazy\"&gt;&lt;/span&gt; é um intervalo e também\n         &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        \\lim_{w \\to b} f^{-1}(w) = f^{-1}(b)\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"1P275\" title=\"1P275\"&gt;\n        &lt;img style=\"height: 3.78ex; vertical-align: -1.76ex; \" src=\"img/img531.svg\" alt=\"$\\displaystyle \\lim_{w \\to b} f^{-1}(w) = f^{-1}(b) $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"1P276\" title=\"1P276\"&gt;\n        para qualquer &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.91ex; vertical-align: -0.36ex; \" src=\"img/img532.svg\" alt=\"$b \\in J$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto é o que afirma o próximo teorema, que enunciaremos sem demonstração. A demonstração pode\n        ser encontrada em [&lt;a href=\"/trigonometria-hiperbolica/referencias#Lima\"&gt;5&lt;/a&gt;, Lima, pág 237].\n        \n        &lt;/p&gt;\n        \n        &lt;div class=\"unidade\" id=\"1Teo11\" title=\"1Teo11\"&gt;&lt;a id=\"teoinvcont\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;1&lt;/span&gt;.&lt;span class=\"arabic\"&gt;11&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n        &lt;i&gt;Se &lt;!-- MATH\n         $I \\subset \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img527.svg\" alt=\"$I \\subset \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; é um intervalo e &lt;!-- MATH\n         $f : I \\to J$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.91ex; vertical-align: -0.36ex; \" src=\"img/img528.svg\" alt=\"$f: I \\to J$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função contínua em todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img533.svg\" alt=\"$a \\in I$\" loading=\"lazy\"&gt;&lt;/span&gt; e que admite função inversa,\n        então a função inversa &lt;!-- MATH\n         $f^{-1}: J \\to I$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.29ex; vertical-align: -0.36ex; \" src=\"img/img534.svg\" alt=\"$f^{-1}: J \\to I$\" loading=\"lazy\"&gt;&lt;/span&gt; é também contínua em &lt;!-- MATH\n         $b = f(a) \\in J$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img535.svg\" alt=\"$b = f(a) \\in J$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/i&gt;&lt;/div&gt;\n        &lt;br /&gt;\n        &lt;p class=\" unidade\" id=\"1P277\" title=\"1P277\"&gt;\n        Este teorema se aplica às seis funções trigonométricas circulares inversas. As seis funções trigonométricas circulares\n        são contínuas em seus respectivos domínios de definição. As restrições bijetivas são todas definidas em conjuntos que\n        são intervalos e, portanto, as funções trigonométricas circulares inversas são contínuas nos seus intervalos de\n        definição, respeitando a lateralidade dos extremos fechados em cada um destes intervalos.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"1P278\" title=\"1P278\"&gt;\n        Resumindo, temos que,\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;table class=\"PAD \" &gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\lim\\limits_{u \\to a} {\\mathrm {sen}}^{-1} u = {\\mathrm {sen}}^{-1} a$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.52ex; vertical-align: -1.60ex; \" src=\"img/img536.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {sen}}^{-1} u = {\\mathrm {sen}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n        &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $a \\in [-1,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img537.svg\" alt=\"$a \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\lim\\limits_{u \\to a} \\cos^{-1} u = \\cos^{-1} a$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.52ex; vertical-align: -1.60ex; \" src=\"img/img538.svg\" alt=\"$\\lim\\limits_{u \\to a} \\cos^{-1} u = \\cos^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n        &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $a \\in [-1,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img537.svg\" alt=\"$a \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\lim\\limits_{u \\to a} {\\mathrm  {tg}}^{-1} u = {\\mathrm  {tg}}^{-1} a$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.52ex; vertical-align: -1.60ex; \" src=\"img/img539.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {tg}}^{-1} u = {\\mathrm {tg}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n        &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $a \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\lim\\limits_{u \\to a} {\\mathrm {ctg}}^{-1} u = {\\mathrm {ctg}}^{-1} a$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.52ex; vertical-align: -1.60ex; \" src=\"img/img540.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {ctg}}^{-1} u = {\\mathrm {ctg}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n        &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $a \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\lim\\limits_{u \\to a} \\sec^{-1} u = \\sec^{-1} a$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.52ex; vertical-align: -1.60ex; \" src=\"img/img541.svg\" alt=\"$\\lim\\limits_{u \\to a} \\sec^{-1} u = \\sec^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n        &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $a \\in (-\\infty,-1] \\cup [1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img542.svg\" alt=\"$a \\in (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; e&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\lim\\limits_{u \\to a} \\csc^{-1} u = \\csc^{-1} a$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.52ex; vertical-align: -1.60ex; \" src=\"img/img543.svg\" alt=\"$\\lim\\limits_{u \\to a} \\csc^{-1} u = \\csc^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n        &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $a \\in (-\\infty,-1] \\cup [1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img542.svg\" alt=\"$a \\in (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"1P279\" title=\"1P279\"&gt;\n        Note que o domínio de definição das funções arco secante e arco cossecante não são intervalos, mas sim uma união de\n        dois intervalos. Estudados separadamente cada um destes intervalos, temos no caso da função arco secante que a função\n        secante é contínua e bijetora de &lt;!-- MATH\n         $(\\frac{\\pi}{2},\\pi]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img544.svg\" alt=\"$(\\frac{\\pi}{2},\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n         $(-\\infty,-1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img545.svg\" alt=\"$(-\\infty,-1]$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, o teorema &lt;a href=\"#teoinvcont\"&gt;1.11&lt;/a&gt; se\n        aplica a este intervalo. Novamente a função secante é contínua e bijetora de &lt;!-- MATH\n         $[0,\\frac{\\pi}{2})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img546.svg\" alt=\"$[0,\\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n         $[1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; e o\n        teorema &lt;a href=\"#teoinvcont\"&gt;1.11&lt;/a&gt; se aplica também a este intervalo. Segue que a função arco secante é contínua em ambos os\n        intervalos &lt;!-- MATH\n         $(-\\infty,-1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img545.svg\" alt=\"$(-\\infty,-1]$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $[1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, contínua na união destes intervalos. Raciocínio similar para a\n        função arco cossecante.\n        \n        &lt;/p&gt;\n        &lt;br&gt;",
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      "Capítulo 1​: Funções trigonométricas circulares​"
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    "title": "Capítulo 1​: Funções trigonométricas circulares​",
    "section": "1.7 Derivadas das funções trigonométricas circulares inversas",
    "text": "1.7 Derivadas das funções trigonométricas circulares inversas\n\n    &lt;p class=\" unidade\" id=\"1P280\" title=\"1P280\"&gt;\n    Estamos agora interessados nas derivadas das funções trigonométricas inversas. Para obter a derivada &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img548.svg\" alt=\"$(f^{-1})'$\" loading=\"lazy\"&gt;&lt;/span&gt; de uma\n    função inversa &lt;!-- MATH\n     $y = f^{-1}(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img549.svg\" alt=\"$y = f^{-1}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, tradicionalmente usamos diferenciação implícita na igualdade &lt;!-- MATH\n     $f(f^{-1}(x)) = x$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img550.svg\" alt=\"$f(f^{-1}(x)) = x$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou\n    equivalentemente, na igualdade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img551.svg\" alt=\"$f(y) = x$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P281\" title=\"1P281\"&gt;\n    Comecemos com a função &lt;!-- MATH\n     $w = f(u) = {\\mathrm {sen}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img426.svg\" alt=\"$w = f(u) = {\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt; definida para todo &lt;!-- MATH\n     $u \\in [-1,1]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;, com &lt;!-- MATH\n     $w \\in\n    [-\\frac{\\pi}{2},\\frac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img424.svg\" alt=\"$w \\in\n    [-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Sabemos que a diferenciação não pode ser estabelecida nos extremos do intervalo\n    fechado e então vamos considerar, ainda bijetivamente, que &lt;!-- MATH\n     $u \\in (-1,1)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img552.svg\" alt=\"$u \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n     $w \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img553.svg\" alt=\"$w \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    Para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; nos intervalos citados, vale a relação &lt;!-- MATH\n     ${\\mathrm {sen}}w =u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img554.svg\" alt=\"${\\mathrm {sen}}w =u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Ao derivarmos com respeito a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, lembremos que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    é variável dependente de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, devemos usar a regra da cadeia. Derivando então a relação &lt;!-- MATH\n     $u = {\\mathrm {sen}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img425.svg\" alt=\"$u = {\\mathrm {sen}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;, com\n    respeito a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    1 = \\frac{d}{du} ({\\mathrm {sen}}w) = \\cos w \\cdot \\frac{dw}{du}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P282\" title=\"1P282\"&gt;\n    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img555.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du} ({\\mathrm {sen}}w) = \\cos w \\cdot \\frac{dw}{du}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P283\" title=\"1P283\"&gt;\n    Como queremos determinar &lt;!-- MATH\n     $w' = \\frac{dw}{du}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img556.svg\" alt=\"$w' = \\frac{dw}{du}$\" loading=\"lazy\"&gt;&lt;/span&gt; vamos então isolar este termo na última igualdade. Obtemos assim,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = \\frac{1}{\\cos w}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P284\" title=\"1P284\"&gt;\n    &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img557.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{1}{\\cos w}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P285\" title=\"1P285\"&gt;\n    Obviamente, queremos também que essa derivada seja dada somente em termos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, e não de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Precisamos substituir a\n    expressão &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img558.svg\" alt=\"$\\cos w$\" loading=\"lazy\"&gt;&lt;/span&gt; do segundo membro, mas só conhecemos a relação &lt;!-- MATH\n     $u = {\\mathrm {sen}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img425.svg\" alt=\"$u = {\\mathrm {sen}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então usamos o fato de que &lt;!-- MATH\n     $\\cos w &gt; 0$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img559.svg\" alt=\"$\\cos w &gt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    para &lt;!-- MATH\n     $w \\in (-\\frac{\\pi}{2},\\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img553.svg\" alt=\"$w \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;, para escrever\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\cos w = \\sqrt{\\cos^{2} w} = \\sqrt{1 - {\\mathrm {sen}}^{2}w} = \\sqrt{1 - u^{2}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P286\" title=\"1P286\"&gt;\n    &lt;img style=\"height: 2.80ex; vertical-align: -0.41ex; \" src=\"img/img560.svg\" alt=\"$\\displaystyle \\cos w = \\sqrt{\\cos^{2} w} = \\sqrt{1 - {\\mathrm {sen}}^{2}w} = \\sqrt{1 - u^{2}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P287\" style=\"text-indent: 0px !important;\" title=\"1P287\"&gt;\n    e segue que\n    &lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{d}{du} {\\mathrm {sen}}^{-1} u = \\frac{dw}{du} = \\frac{1}{\\cos w} = \\frac{1}{\\sqrt{1 - u^{2}}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;/p&gt;&lt;div class=\"mathdisplay unidade\" id=\"1P288\" title=\"1P288\"&gt;\n    &lt;img style=\"height: 5.03ex; vertical-align: -2.06ex; \" src=\"img/img561.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {sen}}^{-1} u = \\frac{dw}{du} = \\frac{1}{\\cos w} = \\frac{1}{\\sqrt{1 - u^{2}}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P289\" style=\"text-indent: 0px !important;\" title=\"1P289\"&gt;\n    para todo &lt;!-- MATH\n     $u \\in (-1,1)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img552.svg\" alt=\"$u \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Note que a derivada não está definida para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img562.svg\" alt=\"$u = -1$\" loading=\"lazy\"&gt;&lt;/span&gt; e para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img309.svg\" alt=\"$u=1$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P290\" title=\"1P290\"&gt;\n    Tomemos agora a função &lt;!-- MATH\n     $w = f(u) = \\cos^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img563.svg\" alt=\"$w = f(u) = \\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida para todo &lt;!-- MATH\n     $u \\in [-1,1]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;, com &lt;!-- MATH\n     $w \\in [0,\\pi]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img564.svg\" alt=\"$w \\in [0,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;. A derivada\n    será estabelecida então para &lt;!-- MATH\n     $u \\in (-1,1)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img552.svg\" alt=\"$u \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n     $w \\in (0,\\pi)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img565.svg\" alt=\"$w \\in (0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então derivamos a relação &lt;!-- MATH\n     $u = \\cos w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img449.svg\" alt=\"$u = \\cos w$\" loading=\"lazy\"&gt;&lt;/span&gt; implicitamente\n    em relação à variável &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtendo\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    1 = \\frac{d}{du}(\\cos w) = -{\\mathrm {sen}}w \\cdot \\frac{dw}{du},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P291\" title=\"1P291\"&gt;\n    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img566.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du}(\\cos w) = -{\\mathrm {sen}}w \\cdot \\frac{dw}{du}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P292\" style=\"text-indent: 0px !important;\" title=\"1P292\"&gt;\n    e isolando o termo de interesse, temos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = - \\frac{1}{{\\mathrm {sen}}w}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P293\" title=\"1P293\"&gt;\n    &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img567.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = - \\frac{1}{{\\mathrm {sen}}w}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P294\" title=\"1P294\"&gt;\n    Novamente, vamos substituir a variável dependente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt;, no segundo membro, pela variável independente. Lembremos que\n    para &lt;!-- MATH\n     $w \\in (0,\\pi)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img565.svg\" alt=\"$w \\in (0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; a função seno é positiva. Segue que\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    {\\mathrm {sen}}w = \\sqrt{{\\mathrm {sen}}^{2} w} = \\sqrt{1-\\cos^{2}w} = \\sqrt{1-u^{2}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P295\" title=\"1P295\"&gt;\n    &lt;img style=\"height: 2.80ex; vertical-align: -0.41ex; \" src=\"img/img568.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}w = \\sqrt{{\\mathrm {sen}}^{2} w} = \\sqrt{1-\\cos^{2}w} = \\sqrt{1-u^{2}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P296\" style=\"text-indent: 0px !important;\" title=\"1P296\"&gt;\n    e, portanto,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{d}{du} \\cos^{-1} u = \\frac{dw}{du} = -\\frac{1}{\\sqrt{1-u^{2}}}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P297\" title=\"1P297\"&gt;\n    &lt;img style=\"height: 5.03ex; vertical-align: -2.06ex; \" src=\"img/img569.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cos^{-1} u = \\frac{dw}{du} = -\\frac{1}{\\sqrt{1-u^{2}}}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P298\" title=\"1P298\"&gt;\n    Tomando agora a função &lt;!-- MATH\n     $w = f(u) = {\\mathrm  {tg}}^{-1}u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img570.svg\" alt=\"$w = f(u) = {\\mathrm {tg}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida para &lt;!-- MATH\n     $u \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, assumindo valores &lt;!-- MATH\n     $w \\in\n    (-\\frac{\\pi}{2},\\frac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img553.svg\" alt=\"$w \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para todo &lt;!-- MATH\n     $u \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos a igualdade &lt;!-- MATH\n     $u = {\\mathrm  {tg}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img571.svg\" alt=\"$u = {\\mathrm {tg}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; e então derivando em relação a\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, vem\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    1 = \\frac{d}{du} {\\mathrm  {tg}}w = \\sec^{2} w \\cdot \\frac{dw}{du}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P299\" title=\"1P299\"&gt;\n    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img572.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du} {\\mathrm {tg}}w = \\sec^{2} w \\cdot \\frac{dw}{du}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P300\" title=\"1P300\"&gt;\n    Desta igualdade e da identidade (&lt;a href=\"#idtgsec\"&gt;1.9&lt;/a&gt;) da proposição &lt;a href=\"#propidsectg\"&gt;1.1&lt;/a&gt;, obtemos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = \\frac{1}{\\sec^{2} w} = \\frac{1}{1 + {\\mathrm  {tg}}^{2}w} = \\frac{1}{1+u^{2}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P301\" title=\"1P301\"&gt;\n    &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img573.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{1}{\\sec^{2} w} = \\frac{1}{1 + {\\mathrm {tg}}^{2}w} = \\frac{1}{1+u^{2}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P302\" style=\"text-indent: 0px !important;\" title=\"1P302\"&gt;\n    e desta forma,\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{d}{du} {\\mathrm  {tg}}^{-1} u = \\frac{dw}{du} = \\frac{1}{1+u^{2}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P303\" title=\"1P303\"&gt;\n    &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img574.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {tg}}^{-1} u = \\frac{dw}{du} = \\frac{1}{1+u^{2}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P304\" style=\"text-indent: 0px !important;\" title=\"1P304\"&gt;\n    para todo &lt;!-- MATH\n     $u \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Note que o lado direito está bem definido para todo &lt;!-- MATH\n     $u \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P305\" title=\"1P305\"&gt;\n    Agora consideremos &lt;!-- MATH\n     $w = f(u) = {\\mathrm {ctg}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img575.svg\" alt=\"$w = f(u) = {\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta função está definida para todo &lt;!-- MATH\n     $u \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, com valores &lt;!-- MATH\n     $w \\in\n    (0,\\pi)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img565.svg\" alt=\"$w \\in (0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Derivando então a igualdade &lt;!-- MATH\n     $u = {\\mathrm {ctg}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img485.svg\" alt=\"$u = {\\mathrm {ctg}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n     &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    1 = \\frac{d}{du} {\\mathrm {ctg}}w = -csc^{2} w \\cdot \\frac{dw}{du},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P306\" title=\"1P306\"&gt;\n    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img576.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du} {\\mathrm {ctg}}w = -csc^{2} w \\cdot \\frac{dw}{du}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P307\" style=\"text-indent: 0px !important;\" title=\"1P307\"&gt;\n    donde temos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = \\frac{-1}{\\csc^{2} w}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P308\" title=\"1P308\"&gt;\n    &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img577.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{-1}{\\csc^{2} w}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P309\" title=\"1P309\"&gt;\n    Usando agora a identidade (&lt;a href=\"#idctgcsc\"&gt;1.8&lt;/a&gt;), da proposição &lt;a href=\"#propidsectg\"&gt;1.1&lt;/a&gt;, temos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = \\frac{-1}{\\csc^{2} w} = \\frac{-1}{{\\mathrm {ctg}}^{2}w + 1} = \\frac{-1}{u^{2}+1},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P310\" title=\"1P310\"&gt;\n    &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img578.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{-1}{\\csc^{2} w} = \\frac{-1}{{\\mathrm {ctg}}^{2}w + 1} = \\frac{-1}{u^{2}+1}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P311\" style=\"text-indent: 0px !important;\" title=\"1P311\"&gt;\n    e obtemos a derivada\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{d}{du} {\\mathrm {ctg}}^{-1} u = -\\frac{1}{1+u^{2}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P312\" title=\"1P312\"&gt;\n    &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img579.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {ctg}}^{-1} u = -\\frac{1}{1+u^{2}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P313\" style=\"text-indent: 0px !important;\" title=\"1P313\"&gt;\n    definida para todo &lt;!-- MATH\n     $u \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P314\" title=\"1P314\"&gt;\n    Seja agora &lt;!-- MATH\n     $w = f(u) = \\sec^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img580.svg\" alt=\"$w = f(u) = \\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida para todo &lt;!-- MATH\n     $u \\in (-\\infty,-1] \\cup [1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img581.svg\" alt=\"$u \\in (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; com valores em &lt;!-- MATH\n     $w \\in [0,\n    \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img582.svg\" alt=\"$w \\in [0,\n    \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para a diferenciação, vamos considerar bijetivamente que &lt;!-- MATH\n     $u \\in\n    (-\\infty,-1) \\cup (1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img583.svg\" alt=\"$u \\in\n    (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; com valores em &lt;!-- MATH\n     $w \\in (0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img584.svg\" alt=\"$w \\in (0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para qualquer &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    no intervalo de diferenciação, temos &lt;!-- MATH\n     $u = \\sec w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img585.svg\" alt=\"$u = \\sec w$\" loading=\"lazy\"&gt;&lt;/span&gt; e derivando esta igualdade em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n     &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    1 = \\frac{d}{du} \\sec w = {\\mathrm  {tg}}w \\sec w \\cdot \\frac{dw}{du},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P315\" title=\"1P315\"&gt;\n    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img586.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du} \\sec w = {\\mathrm {tg}}w \\sec w \\cdot \\frac{dw}{du}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P316\" style=\"text-indent: 0px !important;\" title=\"1P316\"&gt;\n    e isolando o termo de interesse, vem\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = \\frac{1}{{\\mathrm  {tg}}w \\sec w}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P317\" title=\"1P317\"&gt;\n    &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img587.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{1}{{\\mathrm {tg}}w \\sec w}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P318\" title=\"1P318\"&gt;\n    Usaremos novamente a igualdade (&lt;a href=\"#idtgsec\"&gt;1.7&lt;/a&gt;) da proposição &lt;a href=\"#propidsectg\"&gt;1.1&lt;/a&gt;. Extraindo a raiz quadrada em ambos os\n    membros de (&lt;a href=\"#idtgsec\"&gt;1.7&lt;/a&gt;), conseguimos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    |{\\mathrm  {tg}}w| = \\sqrt{\\sec^{2} w - 1}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P319\" title=\"1P319\"&gt;\n    &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img588.svg\" alt=\"$\\displaystyle \\vert{\\mathrm {tg}}w\\vert = \\sqrt{\\sec^{2} w - 1}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P320\" title=\"1P320\"&gt;\n    Como &lt;!-- MATH\n     $w \\in (0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img584.svg\" alt=\"$w \\in (0, \\tfrac{\\pi}{2}) \\cup (\\tfrac{\\pi}{2},\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; não podemos garantir que a tangente de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; seja positiva,\n    mas sim que &lt;!-- MATH\n     ${\\mathrm  {tg}}w \\sec w = \\frac{{\\mathrm {sen}}w}{\\cos w} \\cdot \\frac{1}{\\cos w}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img589.svg\" alt=\"${\\mathrm {tg}}w \\sec w = \\frac{{\\mathrm {sen}}w}{\\cos w} \\cdot \\frac{1}{\\cos w}$\" loading=\"lazy\"&gt;&lt;/span&gt; é positivo. Assim,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P321\" title=\"1P321\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img590.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}w \\sec w$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img591.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm {tg}}w \\sec w\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td&gt;&nbsp;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img592.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm {tg}}w\\vert \\vert\\sec w\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td&gt;&nbsp;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img593.svg\" alt=\"$\\displaystyle = \\vert\\sec w\\vert \\sqrt{\\sec^{2} w - 1} = \\vert u\\vert \\sqrt{u^{2} - 1}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P322\" title=\"1P322\"&gt;\n    Segue que\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{d}{du} \\sec^{-1} u = \\frac{dw}{du} = \\frac{1}{{\\mathrm  {tg}}w \\sec w} = \\frac{1}{|u| \\sqrt{u^{2} - 1}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P323\" title=\"1P323\"&gt;\n    &lt;img style=\"height: 5.35ex; vertical-align: -2.38ex; \" src=\"img/img594.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\sec^{-1} u = \\frac{dw}{du} = \\frac{1}{{\\mathrm {tg}}w \\sec w} = \\frac{1}{\\vert u\\vert \\sqrt{u^{2} - 1}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P324\" style=\"text-indent: 0px !important;\" title=\"1P324\"&gt;\n    para todo &lt;!-- MATH\n     $u \\in (-\\infty,-1) \\cup (1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img583.svg\" alt=\"$u \\in\n    (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P325\" title=\"1P325\"&gt;\n    &lt;b&gt;Nota&lt;/b&gt;: Observe que para tornar a função secante uma função bijetiva, acabamos por escolher um intervalo do domínio onde a\n    função torna-se bijetora. Esta escolha não é única. Outras escolhas também tornam a função secante bijetora. Alguns\n    autores escolhem &lt;!-- MATH\n     $w \\in [0, \\tfrac{\\pi}{2}) \\cup [\\pi, \\tfrac{3\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img595.svg\" alt=\"$w \\in [0, \\tfrac{\\pi}{2}) \\cup [\\pi, \\tfrac{3\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;, pois esta escolha, além de outras\n    implicações, tornará mais simples a fórmula de derivada, que será &lt;!-- MATH\n     $\\frac{d}{du} \\sec^{-1} u = \\frac{1}{u \\sqrt{u^{2} -\n    1}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img596.svg\" alt=\"$\\frac{d}{du} \\sec^{-1} u = \\frac{1}{u \\sqrt{u^{2} -\n    1}}$\" loading=\"lazy\"&gt;&lt;/span&gt;, já que neste intervalo, teríamos &lt;!-- MATH\n     $u = \\sec w \\geq 0$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img597.svg\" alt=\"$u = \\sec w \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, e não precisaríamos manter o módulo. O leitor poderá\n    encontrar em alguns livros de Cálculo Diferencial e Integral esta última fórmula para a derivada da secante inversa.\n    &lt;!-- MATH\n     $\\blacksquare$\n     --&gt;\n    &lt;span style=\"float: right;\" class=\"MATH\"&gt;&lt;img style=\"height: 1.61ex; vertical-align: -0.10ex; \" src=\"img/img279.svg\" alt=\"$\\blacksquare$\" loading=\"lazy\"&gt;&lt;/span&gt;\n     &lt;/p&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P326\" title=\"1P326\"&gt;\n    Para finalizar esta etapa, tomamos &lt;!-- MATH\n     $w = f(u) = \\csc^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img598.svg\" alt=\"$w = f(u) = \\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, que é definida para todo &lt;!-- MATH\n     $u \\in (-\\infty,-1] \\cup [1,\n    \\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img581.svg\" alt=\"$u \\in (-\\infty,-1] \\cup [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n     $w \\in [-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2}]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img599.svg\" alt=\"$w \\in [-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Descartando os extremos fechados de cada intervalo,\n    derivamos &lt;!-- MATH\n     $u = \\csc w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img600.svg\" alt=\"$u = \\csc w$\" loading=\"lazy\"&gt;&lt;/span&gt; em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtendo\n     &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    1 = \\frac{d}{du} \\csc w = -{\\mathrm {ctg}}w \\csc w \\cdot \\frac{dw}{du},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P327\" title=\"1P327\"&gt;\n    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img601.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du} \\csc w = -{\\mathrm {ctg}}w \\csc w \\cdot \\frac{dw}{du}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"1P328\" style=\"text-indent: 0px !important;\" title=\"1P328\"&gt;\n    donde segue\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{dw}{du} = - \\frac{1}{{\\mathrm {ctg}}w \\csc w}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P329\" title=\"1P329\"&gt;\n    &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img602.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = - \\frac{1}{{\\mathrm {ctg}}w \\csc w}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P330\" title=\"1P330\"&gt;\n    Extraindo raiz quadrada em ambos os membros da igualdade (&lt;a href=\"#idctgcsc\"&gt;1.8&lt;/a&gt;), da proposição &lt;a href=\"#propidsectg\"&gt;1.1&lt;/a&gt;, conseguimos\n    a identidade\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    |{\\mathrm {ctg}}w| = \\sqrt{\\csc^{2} w - 1}.\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P331\" title=\"1P331\"&gt;\n    &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img603.svg\" alt=\"$\\displaystyle \\vert{\\mathrm {ctg}}w\\vert = \\sqrt{\\csc^{2} w - 1}. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"1P332\" title=\"1P332\"&gt;\n    Agora como &lt;!-- MATH\n     $w \\in (-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2})$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img604.svg\" alt=\"$w \\in (-\\tfrac{\\pi}{2},0) \\cup (0,\\tfrac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt; não podemos garantir que &lt;!-- MATH\n     ${\\mathrm {ctg}}w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img605.svg\" alt=\"${\\mathrm {ctg}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; seja positiva, mas\n    sabemos que &lt;!-- MATH\n     ${\\mathrm {ctg}}w \\csc w = \\frac{\\cos w}{{\\mathrm {sen}}w} \\cdot \\frac{1}{{\\mathrm {sen}}w}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img606.svg\" alt=\"${\\mathrm {ctg}}w \\csc w = \\frac{\\cos w}{{\\mathrm {sen}}w} \\cdot \\frac{1}{{\\mathrm {sen}}w}$\" loading=\"lazy\"&gt;&lt;/span&gt; é positivo. Então\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P333\" title=\"1P333\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img607.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}w \\csc w$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img608.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm {ctg}}w \\csc w\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td&gt;&nbsp;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img609.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm {ctg}}w\\vert \\vert\\csc w\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td&gt;&nbsp;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img610.svg\" alt=\"$\\displaystyle = \\vert\\csc w\\vert\\sqrt{\\csc^{2} w - 1} = \\vert u\\vert \\sqrt{u^{2}-1},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    &lt;p class=\" unidade\" id=\"1P334\" style=\"text-indent: 0px !important;\" title=\"1P334\"&gt;\n    donde temos\n    &lt;/p&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\frac{d}{du} \\csc^{-1} u = \\frac{dw}{du} = -\\frac{1}{{\\mathrm {ctg}}w \\csc w} = -\\frac{1}{|u| \\sqrt{u^{2}-1}},\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"1P335\" title=\"1P335\"&gt;\n    &lt;img style=\"height: 5.35ex; vertical-align: -2.38ex; \" src=\"img/img611.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\csc^{-1} u = \\frac{dw}{du} = -\\frac{1}{{\\mathrm {ctg}}w \\csc w} = -\\frac{1}{\\vert u\\vert \\sqrt{u^{2}-1}}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    para todo &lt;!-- MATH\n     $u \\in (-\\infty,-1) \\cup (1, \\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img583.svg\" alt=\"$u \\in\n    (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;br /&gt;\n    &lt;p class=\" unidade\" id=\"1P336\" title=\"1P336\"&gt;\n    &lt;b&gt;Nota&lt;/b&gt;: Aqui ocorre o mesmo que o comentado na nota anterior. A escolha de &lt;!-- MATH\n     $w \\in [-\\tfrac{\\pi}{2},0) \\cup [\\tfrac{\\pi}{2},\n    \\pi)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img612.svg\" alt=\"$w \\in [-\\tfrac{\\pi}{2},0) \\cup [\\tfrac{\\pi}{2},\n    \\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;, tornará a fórmula de derivada mais simples. Será &lt;!-- MATH\n     $\\frac{d}{du} \\csc^{-1} u = \\frac{-1}{u \\sqrt{u^{2} - 1}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img613.svg\" alt=\"$\\frac{d}{du} \\csc^{-1} u = \\frac{-1}{u \\sqrt{u^{2} - 1}}$\" loading=\"lazy\"&gt;&lt;/span&gt;, já\n    que no intervalo mencionado temos &lt;!-- MATH\n     $u = \\csc w$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img600.svg\" alt=\"$u = \\csc w$\" loading=\"lazy\"&gt;&lt;/span&gt; sempre positivo. O leitor poderá encontrar em alguns livros de Cálculo\n    Diferencial e Integral esta última fórmula para a derivada da cossecante inversa.\n    &lt;!-- MATH\n     $\\blacksquare$\n     --&gt;\n    &lt;span style=\"float: right;\" class=\"MATH\"&gt;&lt;img style=\"height: 1.61ex; vertical-align: -0.10ex; \" src=\"img/img279.svg\" alt=\"$\\blacksquare$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"1P337\" title=\"1P337\"&gt;\n    Resumimos as derivadas das funções trigonométricas circulares inversas na próxima tabela.\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;&lt;a id=\"1659\"&gt;&lt;/a&gt;\n    &lt;table id=\"1T4\" title=\"1T4\"&gt;\n    &lt;caption&gt;&lt;strong&gt;Tabela 1.4:&lt;/strong&gt;\n    Derivadas das funções trigonométricas circulares inversas.&lt;/caption&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;table class=\"PAD  BORDER\" &gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;derivada&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     ${\\mathrm {sen}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img428.svg\" alt=\"${\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\frac{1}{\\sqrt{1 - u^{2}}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img615.svg\" alt=\"$\\frac{1}{\\sqrt{1 - u^{2}}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n    &lt;br&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\cos^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img521.svg\" alt=\"$\\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $-\\frac{1}{\\sqrt{1 - u^{2}}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img616.svg\" alt=\"$-\\frac{1}{\\sqrt{1 - u^{2}}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n    &lt;br&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     ${\\mathrm  {tg}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img522.svg\" alt=\"${\\mathrm {tg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\frac{1}{1+u^{2}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img617.svg\" alt=\"$\\frac{1}{1+u^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n    &lt;br&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     ${\\mathrm {ctg}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img523.svg\" alt=\"${\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $-\\frac{1}{1+u^{2}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img618.svg\" alt=\"$-\\frac{1}{1+u^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n    &lt;br&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\sec^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img524.svg\" alt=\"$\\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n     $(-\\infty,-1) \\cup (1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $\\frac{1}{|u| \\sqrt{u^{2} - 1}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.54ex; vertical-align: -1.56ex; \" src=\"img/img620.svg\" alt=\"$\\frac{1}{\\vert u\\vert \\sqrt{u^{2} - 1}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n    &lt;br&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\csc^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img525.svg\" alt=\"$\\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n     $(-\\infty,-1) \\cup (1, \\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n     $-\\frac{1}{|u| \\sqrt{u^{2}-1}}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.54ex; vertical-align: -1.56ex; \" src=\"img/img621.svg\" alt=\"$-\\frac{1}{\\vert u\\vert \\sqrt{u^{2}-1}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n    &lt;br&gt;&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P338\" title=\"1P338\"&gt;\n    Observe que as derivadas das inversas das co-funções diferem das derivadas das inversas das funções apenas pelo sinal.\n    Isto é decorrência das relações mencionadas na seção anterior,\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P339\" title=\"1P339\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img622.svg\" alt=\"$\\displaystyle \\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img623.svg\" alt=\"$\\displaystyle = \\tfrac{\\pi}{2} - {\\mathrm {sen}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img624.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img625.svg\" alt=\"$\\displaystyle = \\tfrac{\\pi}{2} - {\\mathrm {tg}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img626.svg\" alt=\"$\\displaystyle \\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img627.svg\" alt=\"$\\displaystyle = \\tfrac{\\pi}{2} - \\sec^{-1} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;p class=\" unidade\" id=\"1P340\" title=\"1P340\"&gt;\n    Derivando estas três igualdades em ambos os membros, com relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"1P341\" title=\"1P341\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img628.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img629.svg\" alt=\"$\\displaystyle = - \\frac{d}{du} {\\mathrm {sen}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img630.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img631.svg\" alt=\"$\\displaystyle = - \\frac{d}{du} {\\mathrm {tg}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img632.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img633.svg\" alt=\"$\\displaystyle = - \\frac{d}{du} \\sec^{-1} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;",
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      "Capítulo 1​: Funções trigonométricas circulares​"
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    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.1 Propriedades da hipérbole",
    "text": "2.1 Propriedades da hipérbole\n\n        &lt;p class=\" unidade\" id=\"2P2\" title=\"2P2\"&gt;\n        Consideremos uma hipérbole de equação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img634.svg\" alt=\"$xy = k$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img635.svg\" alt=\"$k&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para simplificar, vamos considerar que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; são ambos\n        positivos, isto é, estamos tomando apenas um ramo da hipérbole. Os pontos desta curva são da forma &lt;!-- MATH\n         $(x, \\frac{k}{x})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img636.svg\" alt=\"$(x, \\frac{k}{x})$\" loading=\"lazy\"&gt;&lt;/span&gt;\n        para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img637.svg\" alt=\"$x&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt; e o gráfico é a curva da figura &lt;a href=\"#fighip\"&gt;2.1&lt;/a&gt;.\n        &lt;/p&gt;\n        &lt;a id=\"fighip\"&gt;&lt;/a&gt;&lt;a id=\"1702\"&gt;&lt;/a&gt;\n        \n        &lt;div id=\"2I1\" title=\"2I1\" class=\"interativo unidade\"&gt;\n            &lt;div class=\"controles_interatividade\"&gt;\n            \n            &lt;a href=\"interativo/fig2-1.html\" target=\"_blank\" class=\"btn_abrirInterativo\"&gt;Ver maior&lt;/a&gt;&lt;span class=\"barra\"&gt; | &lt;/span&gt;&lt;span class=\"referencia\" onclick=\"alert('A referência é: 2I1.')\"&gt;Referência&lt;/span&gt;\n            \n            &lt;/div&gt;\n            \n            &lt;iframe id=\"fig2-1\" class=\"graficos\" loading=\"lazy\" src=\"interativo/fig2-1.html\"&gt;&lt;/iframe&gt;\n            &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P3\" title=\"2P3\"&gt;\n        Dado um número real &lt;!-- MATH\n         $\\alpha &gt; 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img638.svg\" alt=\"$\\alpha &gt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, vamos considerar a transformação\n        &lt;/p&gt;\n         &lt;br&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P4\" title=\"2P4\"&gt;&lt;a id=\"TransfT\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{eqnarray}\n        T : \\mathbb{R}^{2} & \\to & \\mathbb{R}^{2} \\nonumber \\\\\n        (x,y) & \\mapsto & T(x,y) = (\\alpha x, \\tfrac{1}{\\alpha} y).\n        \\end{eqnarray}\n         --&gt;\n        &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n        &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.10ex; \" src=\"img/img639.svg\" alt=\"$\\displaystyle T : \\mathbb{R}^{2}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n        &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n        &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.10ex; \" src=\"img/img640.svg\" alt=\"$\\displaystyle \\mathbb{R}^{2}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n        &nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img641.svg\" alt=\"$\\displaystyle (x,y)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n        &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n        &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img642.svg\" alt=\"$\\displaystyle T(x,y) = (\\alpha x, \\tfrac{1}{\\alpha} y).$\" loading=\"lazy\"&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n        (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;br clear=\"ALL\"&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P5\" title=\"2P5\"&gt;\n        Esta transformação é conhecida como deslocamento ou deslizamento sobre a hipérbole. Isto se deve ao fato de que\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; leva pontos da hipérbole na hipérbole (ver proposição &lt;a href=\"#prophip\"&gt;2.1&lt;/a&gt;). Esta transformação é bastante importante no\n        nosso estudo e possui propriedades interessantes. As próximas proposições evidenciam algumas destas propriedades.\n        Outras propriedades podem ser encontradas em [&lt;a href=\"/trigonometria-hiperbolica/referencias#Shervatov\"&gt;7&lt;/a&gt;, Shervatov].\n        \n        &lt;/p&gt;\n        \n        &lt;div id=\"2Teo1\" title=\"2Teo1\" class=\" unidade\"&gt;&lt;a id=\"prophip\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; \n        &lt;i&gt;A transformação &lt;!-- MATH\n         $T: \\mathbb{R}^{2} \\to \\mathbb{R}^{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img644.svg\" alt=\"$T: \\mathbb{R}^{2} \\to \\mathbb{R}^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida por &lt;/i&gt;(&lt;a href=\"#TransfT\"&gt;2.1&lt;/a&gt;)&lt;i&gt;, possui as seguintes propriedades: \n        &lt;br&gt;\n        &lt;!-- MATH\n         $\\mathbf{(a)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img645.svg\" alt=\"$\\mathbf{(a)}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; leva pontos da hipérbole em pontos da hipérbole. \n        &lt;br&gt;\n        &lt;!-- MATH\n         $\\mathbf{(b)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img646.svg\" alt=\"$\\mathbf{(b)}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; leva retas do plano em retas. \n        &lt;br&gt;\n        &lt;!-- MATH\n         $\\mathbf{(c)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img647.svg\" alt=\"$\\mathbf{(c)}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; preserva a razão entre os comprimentos de segmentos de uma mesma reta. \n        &lt;br&gt;\n        &lt;!-- MATH\n         $\\mathbf{(d)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img648.svg\" alt=\"$\\mathbf{(d)}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; leva retas paralelas em retas paralelas. \n        &lt;br&gt;\n        &lt;!-- MATH\n         $\\mathbf{(e)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img649.svg\" alt=\"$\\mathbf{(e)}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; preserva as assíntotas da hipérbole.\n        &lt;/i&gt;&lt;/div&gt;\n        &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n        \n        Dado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img650.svg\" alt=\"$a &gt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, o ponto &lt;!-- MATH\n         $A = (a,\\frac{k}{a})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img651.svg\" alt=\"$A = (a,\\frac{k}{a})$\" loading=\"lazy\"&gt;&lt;/span&gt; pertencente à hipérbole e temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        T(A) = T(a, \\tfrac{k}{a}) = (\\alpha a, \\tfrac{1}{\\alpha} \\tfrac{k}{a}) = (\\alpha a, \\tfrac{k}{\\alpha a}),\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P6\" title=\"2P6\"&gt;\n        &lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img652.svg\" alt=\"$\\displaystyle T(A) = T(a, \\tfrac{k}{a}) = (\\alpha a, \\tfrac{1}{\\alpha} \\tfrac{k}{a}) = (\\alpha a, \\tfrac{k}{\\alpha a}), $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        sendo que claramente o membro da direita é um ponto que ainda está sobre a hipérbole, uma vez que satisfaz a equação\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img634.svg\" alt=\"$xy = k$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto prova &lt;!-- MATH\n         $\\mathbf{(a)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img645.svg\" alt=\"$\\mathbf{(a)}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para provar &lt;!-- MATH\n         $\\mathbf{(b)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img646.svg\" alt=\"$\\mathbf{(b)}$\" loading=\"lazy\"&gt;&lt;/span&gt; consideremos a parametrização &lt;!-- MATH\n         $(x,y) = (c,d) +\n        t(a,b) = (at+c,bt+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img653.svg\" alt=\"$(x,y) = (c,d) +\n        t(a,b) = (at+c,bt+d)$\" loading=\"lazy\"&gt;&lt;/span&gt; que descreve os pontos de uma reta, fazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; variar em &lt;!-- MATH\n         $\\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, para quaisquer &lt;!-- MATH\n         $a,b,c,d \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img654.svg\" alt=\"$a,b,c,d \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        desde que &lt;!-- MATH\n         $(a,b) \\neq (0,0)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img655.svg\" alt=\"$(a,b) \\neq (0,0)$\" loading=\"lazy\"&gt;&lt;/span&gt; pois é o vetor diretor da reta. Então aplicando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        T(at+c,bt+d) = (\\alpha at + \\alpha c, \\tfrac{bt+d}{\\alpha}) = ( \\alpha at + \\alpha c, \\tfrac{b}{\\alpha} t + \\tfrac{d}{\\alpha}),\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P7\" title=\"2P7\"&gt;\n        &lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img656.svg\" alt=\"$\\displaystyle T(at+c,bt+d) = (\\alpha at + \\alpha c, \\tfrac{bt+d}{\\alpha}) = ( \\alpha at + \\alpha c, \\tfrac{b}{\\alpha} t + \\tfrac{d}{\\alpha}), $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        e obviamente os pontos do membro da direita descrevem uma reta fazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; variar em &lt;!-- MATH\n         $\\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para o item &lt;!-- MATH\n         $\\mathbf{(c)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img647.svg\" alt=\"$\\mathbf{(c)}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        consideremos novamente a parametrização &lt;!-- MATH\n         $(at+c,bt+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img657.svg\" alt=\"$(at+c,bt+d)$\" loading=\"lazy\"&gt;&lt;/span&gt; de uma reta qualquer. O comprimento de um segmento\n        &lt;!-- MATH\n         $\\overline{AB}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.07ex; vertical-align: -0.10ex; \" src=\"img/img658.svg\" alt=\"$\\overline{AB}$\" loading=\"lazy\"&gt;&lt;/span&gt; é a distância entre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se &lt;!-- MATH\n         $A = (at_{1}+c, bt_{1}+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img659.svg\" alt=\"$A = (at_{1}+c, bt_{1}+d)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $B = (at_{2}+c, bt_{2}+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img660.svg\" alt=\"$B = (at_{2}+c, bt_{2}+d)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P8\" title=\"2P8\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.07ex; vertical-align: -0.10ex; \" src=\"img/img661.svg\" alt=\"$\\displaystyle \\overline{AB}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img662.svg\" alt=\"$\\displaystyle = \\vert B-A\\vert = \\vert(a(t_{2}-t_{1}), b(t_{2}-t_{1}))\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.07ex; vertical-align: -1.22ex; \" src=\"img/img663.svg\" alt=\"$\\displaystyle = \\sqrt{a^{2}(t_{2}-t_{1})^{2} + b^{2}(t_{2}-t_{1})^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img664.svg\" alt=\"$\\displaystyle = \\vert t_{2}-t_{1}\\vert\\sqrt{a^{2} + b^{2}}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"2P9\" title=\"2P9\"&gt;\n        Se &lt;!-- MATH\n         $\\overline{CD}$\n         --&gt;\n        \n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.07ex; vertical-align: -0.10ex; \" src=\"img/img665.svg\" alt=\"$\\overline{CD}$\" loading=\"lazy\"&gt;&lt;/span&gt; é outro segmento desta reta com &lt;!-- MATH\n         $C = (at_{3}+c, bt_{3}+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img666.svg\" alt=\"$C = (at_{3}+c, bt_{3}+d)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $D = (at_{4}+c, bt_{4}+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img667.svg\" alt=\"$D = (at_{4}+c, bt_{4}+d)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então da\n        mesma forma, o comprimento do segmento &lt;!-- MATH\n         $\\overline{CD}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.07ex; vertical-align: -0.10ex; \" src=\"img/img665.svg\" alt=\"$\\overline{CD}$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual a &lt;!-- MATH\n         $|t_{4}-t_{3}|\\sqrt{a^{2} + b^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img668.svg\" alt=\"$\\vert t_{4}-t_{3}\\vert\\sqrt{a^{2} + b^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; e a razão entre os\n        segmentos é\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{\\overline{CD}}{\\overline{AB}}\n         = \\frac{|t_{4}-t_{3}|\\sqrt{a^{2} + b^{2}}}{|t_{2}-t_{1}|\\sqrt{a^{2} + b^{2}}} = \\frac{|t_{4}-t_{3}|}{|t_{2}-t_{1}|}, \n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P10\" title=\"2P10\"&gt;\n        &lt;img style=\"height: 5.87ex; vertical-align: -2.38ex; \" src=\"img/img669.svg\" alt=\"$\\displaystyle \\frac{\\overline{CD}}{\\overline{AB}}\n        = \\frac{\\vert t_{4}-t_{3}\\v...\n        ...\\sqrt{a^{2} + b^{2}}} = \\frac{\\vert t_{4}-t_{3}\\vert}{\\vert t_{2}-t_{1}\\vert}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        pois &lt;!-- MATH\n         $\\sqrt{a^{2}+b^{2}} = 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img670.svg\" alt=\"$\\sqrt{a^{2}+b^{2}} = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P11\" title=\"2P11\"&gt;\n        Vamos então provar que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img671.svg\" alt=\"$A' = T(A)$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img672.svg\" alt=\"$B' = T(B)$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img673.svg\" alt=\"$C' = T(C)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img674.svg\" alt=\"$D' = T(D)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então\n        &lt;!-- MATH\n         $\\frac{\\overline{C'D'}}{\\overline{A'B'}} = \\frac{|t_{4}-t_{3}|}{|t_{2}-t_{1}|}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.58ex; vertical-align: -1.24ex; \" src=\"img/img675.svg\" alt=\"$\\frac{\\overline{C'D'}}{\\overline{A'B'}} = \\frac{\\vert t_{4}-t_{3}\\vert}{\\vert t_{2}-t_{1}\\vert}$\" loading=\"lazy\"&gt;&lt;/span&gt;. De fato, &lt;!-- MATH\n         $A' = (\\alpha at_{1} + \\alpha\n        c, \\frac{b}{\\alpha}t_{1} + \\frac{d}{\\alpha})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img676.svg\" alt=\"$A' = (\\alpha at_{1} + \\alpha\n        c, \\frac{b}{\\alpha}t_{1} + \\frac{d}{\\alpha})$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;!-- MATH\n         $B' = (\\alpha at_{2} + \\alpha c, \\frac{b}{\\alpha}t_{2} +\n        \\frac{d}{\\alpha})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img677.svg\" alt=\"$B' = (\\alpha at_{2} + \\alpha c, \\frac{b}{\\alpha}t_{2} +\n        \\frac{d}{\\alpha})$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;!-- MATH\n         $C' = (\\alpha at_{3} + \\alpha c, \\frac{b}{\\alpha}t_{3} + \\frac{d}{\\alpha})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img678.svg\" alt=\"$C' = (\\alpha at_{3} + \\alpha c, \\frac{b}{\\alpha}t_{3} + \\frac{d}{\\alpha})$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $D' = (\\alpha at_{4}\n        + \\alpha c, \\frac{b}{\\alpha}t_{4} + \\frac{d}{\\alpha})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img679.svg\" alt=\"$D' = (\\alpha at_{4}\n        + \\alpha c, \\frac{b}{\\alpha}t_{4} + \\frac{d}{\\alpha})$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P12\" title=\"2P12\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.08ex; vertical-align: -0.10ex; \" src=\"img/img680.svg\" alt=\"$\\displaystyle \\overline{A'B'}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img681.svg\" alt=\"$\\displaystyle = \\vert B' - A'\\vert = \\vert(\\alpha a(t_{2}-t_{1}), \\frac{b}{\\alpha}(t_{2}-t_{1}))\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.33ex; vertical-align: -1.60ex; \" src=\"img/img682.svg\" alt=\"$\\displaystyle = \\sqrt{\\alpha^{2}a^{2}(t_{2}-t_{1})^{2} + \\frac{b^{2}}{\\alpha^{2}}(t_{2}-t_{1})^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.33ex; vertical-align: -1.60ex; \" src=\"img/img683.svg\" alt=\"$\\displaystyle = \\vert t_{2}-t_{1}\\vert\\sqrt{\\alpha^{2}a^{2} + \\frac{b^{2}}{\\alpha^{2}}}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p style=\"text-indent:0px !important;\" class=\" unidade\" id=\"2P13\" title=\"2P13\"&gt;\n        e da mesma forma\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        \\overline{C'D'} = |t_{4}-t_{3}|\\sqrt{\\alpha^{2}a^{2} + \\frac{b^{2}}{\\alpha^{2}}},\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P14\" title=\"2P14\"&gt;\n        &lt;img style=\"height: 5.33ex; vertical-align: -1.60ex; \" src=\"img/img684.svg\" alt=\"$\\displaystyle \\overline{C'D'} = \\vert t_{4}-t_{3}\\vert\\sqrt{\\alpha^{2}a^{2} + \\frac{b^{2}}{\\alpha^{2}}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p style=\"text-indent:0px !important;\" class=\" unidade\" id=\"2P15\" title=\"2P15\"&gt;\n        e, portanto,\n        &lt;/p&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{\\overline{C'D'}}{\\overline{A'B'}}\n         = \\frac{|t_{4}-t_{3}|\\sqrt{\\alpha^{2}a^{2} + \\frac{b^{2}}{\\alpha^{2}}}}{|t_{2}-t_{1}|\\sqrt{\\alpha^{2}a^{2} + \\frac{b^{2}}{\\alpha^{2}}}} = \\frac{|t_{4}-t_{3}|}{|t_{2}-t_{1}|}\n         = \\frac{\\overline{CD}}{\\overline{AB}}, \n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P16\" title=\"2P16\"&gt;\n        &lt;img style=\"height: 8.53ex; vertical-align: -3.75ex; \" src=\"img/img685.svg\" alt=\"$\\displaystyle \\frac{\\overline{C'D'}}{\\overline{A'B'}}\n        = \\frac{\\vert t_{4}-t_{...\n        ...}-t_{3}\\vert}{\\vert t_{2}-t_{1}\\vert}\n        = \\frac{\\overline{CD}}{\\overline{AB}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        &lt;p style=\"text-indent:0px !important;\" class=\" unidade\" id=\"2P17\" title=\"2P17\"&gt;e isso prova o item &lt;!-- MATH\n         $\\mathbf{(c)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img647.svg\" alt=\"$\\mathbf{(c)}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P18\" title=\"2P18\"&gt;\n        Para provar &lt;!-- MATH\n         $\\mathbf{(d)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img648.svg\" alt=\"$\\mathbf{(d)}$\" loading=\"lazy\"&gt;&lt;/span&gt;, tomemos duas retas paralelas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, os coeficientes são proporcionais. Tomemos\n        as parametrizações &lt;!-- MATH\n         $r:(at+c,bt+d)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img686.svg\" alt=\"$r:(at+c,bt+d)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $s:( (ma)t + e, (mb)t + f)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img687.svg\" alt=\"$s:( (ma)t + e, (mb)t + f)$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n         $t \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img688.svg\" alt=\"$t \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então, &lt;!-- MATH\n         $T(r) = ( (\\alpha a) t +\n        \\alpha c, \\frac{b}{\\alpha}t + \\frac{d}{\\alpha})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img689.svg\" alt=\"$T(r) = ( (\\alpha a) t +\n        \\alpha c, \\frac{b}{\\alpha}t + \\frac{d}{\\alpha})$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $T(s) = ( (\\alpha ma) t + \\alpha e, \\frac{mb}{\\alpha}t +\n        \\frac{f}{\\alpha})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img690.svg\" alt=\"$T(s) = ( (\\alpha ma) t + \\alpha e, \\frac{mb}{\\alpha}t +\n        \\frac{f}{\\alpha})$\" loading=\"lazy\"&gt;&lt;/span&gt; e, claramente, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img691.svg\" alt=\"$T(r)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img692.svg\" alt=\"$T(s)$\" loading=\"lazy\"&gt;&lt;/span&gt; são paralelas, já que os seus coeficientes são proporcionais.\n        &lt;/p&gt;\n        &lt;p&gt;\n        Finalmente vamos a &lt;!-- MATH\n         $\\mathbf{(e)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img649.svg\" alt=\"$\\mathbf{(e)}$\" loading=\"lazy\"&gt;&lt;/span&gt;. As assíntotas da hipérbole são os eixos coordenados. As suas parametrizações são\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img693.svg\" alt=\"$(t,0)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img694.svg\" alt=\"$(0,t)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Obviamente &lt;!-- MATH\n         $T(t,0) = (\\alpha t,0)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img695.svg\" alt=\"$T(t,0) = (\\alpha t,0)$\" loading=\"lazy\"&gt;&lt;/span&gt;, que continua sendo o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $T(0,t) = (0, \\frac{1}{\\alpha}\n        t)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img696.svg\" alt=\"$T(0,t) = (0, \\frac{1}{\\alpha}\n        t)$\" loading=\"lazy\"&gt;&lt;/span&gt;, que continua sendo o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n        &lt;/p&gt;\n        &lt;/div&gt;\n    \n        &lt;p class=\" unidade\" id=\"2P19\" title=\"2P19\"&gt;\n        &lt;b&gt;Nota&lt;/b&gt;: Observe que, na transformação dada em (&lt;a href=\"#TransfT\"&gt;2.1&lt;/a&gt;), para &lt;!-- MATH\n         $\\alpha = 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img697.svg\" alt=\"$\\alpha = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; temos a aplicação identidade. Para &lt;!-- MATH\n         $\\alpha &gt;\n        1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img698.svg\" alt=\"$\\alpha &gt;\n        1$\" loading=\"lazy\"&gt;&lt;/span&gt;, um dado ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; será deslocado por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; no sentido do crescimento do eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. E se &lt;!-- MATH\n         $0 &lt; \\alpha &lt; 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img699.svg\" alt=\"$0 &lt; \\alpha &lt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, então o\n        deslocamento de um determinado ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; da hipérbole, se dará no sentido contrário ao do crescimento do eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;!-- MATH\n         $\\blacksquare$\n         --&gt;\n        &lt;span style=\"float: right\" class=\"MATH\"&gt;&lt;img style=\"height: 1.61ex; vertical-align: -0.10ex; \" src=\"img/img279.svg\" alt=\"$\\blacksquare$\" loading=\"lazy\"&gt;&lt;/span&gt;\n         &lt;/p&gt;\n        &lt;br&gt;\n       \n        &lt;div id=\"2Teo2\" title=\"2Teo2\" class=\" unidade\"&gt;&lt;a id=\"nochangearea\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n        &lt;i&gt;A transformação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; não altera a área de figuras do plano.\n        &lt;/i&gt;&lt;/div&gt;\n        \n      \n        &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n        \n        De fato, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; pode ser escrita como composição das transformações\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\begin{array}{rcl}\n        T_{1} : \\mathbb{R}^{2} & \\to & \\mathbb{R}^{2} \\\\\n        (x,y) & \\mapsto & (\\alpha x, y)\n        \\end{array} \\qquad \\qquad \\text{e} \\qquad \\qquad\n        \\begin{array}{rcl}\n        T_{2} : \\mathbb{R}^{2} & \\to & \\mathbb{R}^{2} \\\\\n        (x,y) & \\mapsto & (x, \\tfrac{1}{\\alpha}y).\n        \\end{array}\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"2P20\" title=\"2P20\"&gt;\n        &lt;img style=\"height: 6.47ex; vertical-align: -2.72ex; \" src=\"img/img700.svg\" alt=\"$\\displaystyle \\begin{array}{rcl}\n        T_{1} : \\mathbb{R}^{2} &amp; \\to &amp; \\mathbb{R}^{2} \\\\\n        (x,y) &amp; \\mapsto &amp; (\\alpha x, y)\n        \\end{array}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img src=\"img/img701.svg\" alt=\"\\begin{displaymath}\\qquad \\qquad\n        \\begin{array}{rcl}\n        T_{2} : \\mathbb{R}^{2} &amp; \\...\n        ...2} \\\\\n        (x,y) &amp; \\mapsto &amp; (x, \\tfrac{1}{\\alpha}y).\n        \\end{array}\\end{displaymath}\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P21\" title=\"2P21\"&gt;\n        Sabemos da geometria plana que a transformação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img702.svg\" alt=\"$T_{1}$\" loading=\"lazy\"&gt;&lt;/span&gt; altera a área de uma figura plana multiplicando esta área por\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; e a segunda transformação multiplica a área de uma figura por &lt;!-- MATH\n         $\\frac{1}{\\alpha}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img703.svg\" alt=\"$\\frac{1}{\\alpha}$\" loading=\"lazy\"&gt;&lt;/span&gt;. A composta das duas\n        aplicações então multiplica a área de figuras primeiro por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img13.svg\" alt=\"$\\alpha$\" loading=\"lazy\"&gt;&lt;/span&gt; e depois por &lt;!-- MATH\n         $\\frac{1}{\\alpha}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img703.svg\" alt=\"$\\frac{1}{\\alpha}$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, não\n        altera a área de figuras planas.\n        &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n        &lt;/p&gt;&lt;/div&gt;&lt;p class=\" unidade\" id=\"2P22\" title=\"2P22\"&gt;&lt;/p&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P23\" title=\"2P23\"&gt;\n        A figura &lt;a href=\"#fighiptrig\"&gt;2.2&lt;/a&gt; representa um braço da hipérbole de equação cartesiana &lt;!-- MATH\n         $x^{2} - y^{2} = 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img704.svg\" alt=\"$x^{2} - y^{2} = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta “metade”\n        de hipérbole é conhecida como hipérbole trigonométrica. &lt;a name=\"1922\"&gt;&lt;/a&gt;\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P24\" title=\"2P24\"&gt;\n        Note que, se rotacionarmos o gráfico da figura &lt;a href=\"#fighiptrig\"&gt;2.2&lt;/a&gt;, &lt;!-- MATH\n         $45^{\\circ}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.13ex; \" src=\"img/img705.svg\" alt=\"$45^{\\circ}$\" loading=\"lazy\"&gt;&lt;/span&gt; no sentido anti-horário com relação a\n        origem, este braço de hipérbole se torna o braço de hipérbole da figura &lt;a href=\"#fighip\"&gt;2.1&lt;/a&gt;, bastando apenas ajustar o valor\n        de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também as assíntotas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img706.svg\" alt=\"$y = \\pm x$\" loading=\"lazy\"&gt;&lt;/span&gt;, após esta rotação, se tornam os eixos coordenados da figura &lt;a href=\"#fighip\"&gt;2.1&lt;/a&gt;.\n        Isso significa que, por uma rotação, as propriedades listadas nas Proposições &lt;a href=\"#prophip\"&gt;2.1&lt;/a&gt; e &lt;a href=\"#nochangearea\"&gt;2.2&lt;/a&gt; são\n        válidas também na hipérbole trigonométrica e suas assíntotas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img706.svg\" alt=\"$y = \\pm x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto porque a rotação (de &lt;!-- MATH\n         $45^{\\circ}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.13ex; \" src=\"img/img705.svg\" alt=\"$45^{\\circ}$\" loading=\"lazy\"&gt;&lt;/span&gt;), é\n        um movimento rígido e preserva comprimento de segmentos, relação de paralelismo e medidas de áreas.\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"fighiptrig\"&gt;&lt;/a&gt;&lt;a id=\"1932\"&gt;&lt;/a&gt;\n        &lt;table id=\"2I2\" title=\"2I2\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.2:&lt;/strong&gt;\n        Hipérbole trigonométrica.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/hiptrig.png\" alt=\"Image hiptrig\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P25\" title=\"2P25\"&gt;\n        Chamemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img34.svg\" alt=\"$V=(1,0)$\" loading=\"lazy\"&gt;&lt;/span&gt; o vértice da hipérbole da figura &lt;a href=\"#fighiptrig\"&gt;2.2&lt;/a&gt;. Consideremos um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre a hipérbole\n        situado no primeiro quadrante (fig. &lt;a href=\"#figtrig1\"&gt;2.3&lt;/a&gt;). Pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; traçamos a perpendicular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img68.svg\" alt=\"$AP$\" loading=\"lazy\"&gt;&lt;/span&gt; ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Marcamos\n        o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; simétrico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. O segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img707.svg\" alt=\"$AB$\" loading=\"lazy\"&gt;&lt;/span&gt; é dito segmento conjugado do segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt;, pois\n        o prolongamento de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt; encontra &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img707.svg\" alt=\"$AB$\" loading=\"lazy\"&gt;&lt;/span&gt; no ponto médio &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt; de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img707.svg\" alt=\"$AB$\" loading=\"lazy\"&gt;&lt;/span&gt;. O ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; está então sobre a hipérbole no quarto\n        quadrante e o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img707.svg\" alt=\"$AB$\" loading=\"lazy\"&gt;&lt;/span&gt; é também perpendicular ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Traçamos pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; paralela a assíntota\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img708.svg\" alt=\"$y=x$\" loading=\"lazy\"&gt;&lt;/span&gt; e pelo ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; a reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt; paralela a assíntota &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img709.svg\" alt=\"$y=-x$\" loading=\"lazy\"&gt;&lt;/span&gt;. As retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt; se encontram no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt;, sobre o\n        eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;, formando o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img710.svg\" alt=\"$AMB$\" loading=\"lazy\"&gt;&lt;/span&gt; retângulo em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"figtrig1\"&gt;&lt;/a&gt;&lt;a id=\"1939\"&gt;&lt;/a&gt;\n        &lt;table id=\"2I3\" title=\"2I3\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.3:&lt;/strong&gt;\n         Construindo propriedades adicionais na hipérbole. &lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/fighip01.png\" alt=\"Image fighip01\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P26\" title=\"2P26\"&gt;\n        Aplicamos agora um deslizamento sobre a hipérbole, isto é, aplicamos a transformação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; dada em (&lt;a href=\"#TransfT\"&gt;2.1&lt;/a&gt;).\n        Obtemos assim os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img711.svg\" alt=\"$B'$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img712.svg\" alt=\"$P'$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img713.svg\" alt=\"$V'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img714.svg\" alt=\"$M'$\" loading=\"lazy\"&gt;&lt;/span&gt; imagem pela &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; dos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img310.svg\" alt=\"$V$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt;\n        respectivamente; e as retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img715.svg\" alt=\"$r'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img716.svg\" alt=\"$s'$\" loading=\"lazy\"&gt;&lt;/span&gt; imagem pela &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt; das retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt; respectivamente, conforme figura\n        &lt;a href=\"#figtrig2\"&gt;2.4&lt;/a&gt;.\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"figtrig2\"&gt;&lt;/a&gt;&lt;a id=\"1946\"&gt;&lt;/a&gt;\n        &lt;table id=\"2I4\" title=\"2I4\"&gt;\n        &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.4:&lt;/strong&gt;\n         Propriedades adicionais na hipérbole. &lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;img src=\"img/fighip02.png\" alt=\"Image fighip02\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P27\" title=\"2P27\"&gt;\n        Nestes termos, como os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt; estão sobre uma mesma reta, pela propriedade &lt;!-- MATH\n         $\\mathbf{(b)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img646.svg\" alt=\"$\\mathbf{(b)}$\" loading=\"lazy\"&gt;&lt;/span&gt; da Proposição\n        &lt;a href=\"#prophip\"&gt;2.1&lt;/a&gt; os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img711.svg\" alt=\"$B'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img712.svg\" alt=\"$P'$\" loading=\"lazy\"&gt;&lt;/span&gt; estão também sobre uma mesma reta. Pela mesma razão, os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img717.svg\" alt=\"$O$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img714.svg\" alt=\"$M'$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img713.svg\" alt=\"$V'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img712.svg\" alt=\"$P'$\" loading=\"lazy\"&gt;&lt;/span&gt; também estão alinhados, isto é, sobre uma reta. Os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img713.svg\" alt=\"$V'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img711.svg\" alt=\"$B'$\" loading=\"lazy\"&gt;&lt;/span&gt; ainda estão sobre a hipérbole.\n        A razão entre as medidas dos segmentos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img68.svg\" alt=\"$AP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img718.svg\" alt=\"$PB$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual 1 e, portanto, pelo item &lt;!-- MATH\n         $\\mathbf{(c)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img647.svg\" alt=\"$\\mathbf{(c)}$\" loading=\"lazy\"&gt;&lt;/span&gt; da Proposição\n        &lt;a href=\"#prophip\"&gt;2.1&lt;/a&gt;, a razão entre as medidas dos segmentos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img719.svg\" alt=\"$A'P'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img720.svg\" alt=\"$P'B'$\" loading=\"lazy\"&gt;&lt;/span&gt; é também 1, isto é, o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img712.svg\" alt=\"$P'$\" loading=\"lazy\"&gt;&lt;/span&gt; é ainda o ponto\n        médio do segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img721.svg\" alt=\"$A'B'$\" loading=\"lazy\"&gt;&lt;/span&gt;. Pelas propriedades &lt;!-- MATH\n         $\\mathbf{(d)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img648.svg\" alt=\"$\\mathbf{(d)}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $\\mathbf{(e)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img649.svg\" alt=\"$\\mathbf{(e)}$\" loading=\"lazy\"&gt;&lt;/span&gt; da mesma Proposição, as retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img715.svg\" alt=\"$r'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img716.svg\" alt=\"$s'$\" loading=\"lazy\"&gt;&lt;/span&gt;\n        são paralelas as retas imagens das assíntotas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img706.svg\" alt=\"$y = \\pm x$\" loading=\"lazy\"&gt;&lt;/span&gt; por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como as assíntotas não são alteradas, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img715.svg\" alt=\"$r'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img716.svg\" alt=\"$s'$\" loading=\"lazy\"&gt;&lt;/span&gt;\n        ainda são paralelas as assíntotas. Isto significa que o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img722.svg\" alt=\"$A'B'M'$\" loading=\"lazy\"&gt;&lt;/span&gt; é ainda um triângulo retângulo em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img714.svg\" alt=\"$M'$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n         &lt;/p&gt;",
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      "Capítulo 2: Funções trigonométricas hiperbólicas"
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    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.2 A trigonometria hiperbólica",
    "text": "2.2 A trigonometria hiperbólica\n\n            &lt;p class=\" unidade\" id=\"2P28\" title=\"2P28\"&gt;\n            A trigonometria hiperbólica é construída sobre a hipérbole trigonométrica, isto é, o braço da hipérbole de equação\n            &lt;!-- MATH\n             $x^{2} - y^{2} = 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img704.svg\" alt=\"$x^{2} - y^{2} = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, representado na figura &lt;a href=\"#fighiptrig\"&gt;2.2&lt;/a&gt;. Dado um número real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img46.svg\" alt=\"$u \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, entenderemos por ângulo\n            hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img723.svg\" alt=\"$VOA$\" loading=\"lazy\"&gt;&lt;/span&gt;, o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; da hipérbole no primeiro quadrante, de forma que a área do\n            setor &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img60.svg\" alt=\"$OVA$\" loading=\"lazy\"&gt;&lt;/span&gt; seja igual a &lt;!-- MATH\n             $\\frac{u}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img61.svg\" alt=\"$\\frac{u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; (Ver figura &lt;a href=\"#figanghippos\"&gt;2.5&lt;/a&gt;). No caso em que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img52.svg\" alt=\"$u &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt; o ângulo hiperbólico\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; é o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; da hipérbole, no quarto quadrante, de forma que a área do setor &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img60.svg\" alt=\"$OVA$\" loading=\"lazy\"&gt;&lt;/span&gt; seja igual a &lt;!-- MATH\n             $\\frac{-u}{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.83ex; \" src=\"img/img724.svg\" alt=\"$\\frac{-u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;a name=\"1964\"&gt;&lt;/a&gt;&lt;/p&gt;\n            \n            &lt;a id=\"figanghippos\"&gt;&lt;/a&gt;&lt;a id=\"1967\"&gt;&lt;/a&gt;\n                &lt;div id=\"2I5\" title=\"2I5\" class=\"interativo unidade\"&gt;\n                    &lt;div class=\"controles_interatividade\"&gt;\n                    \n                    &lt;a href=\"interativo/fig2-5.html\" target=\"_blank\" class=\"btn_abrirInterativo\"&gt;Ver maior&lt;/a&gt;&lt;span class=\"barra\"&gt; | &lt;/span&gt;&lt;span class=\"referencia\" onclick=\"alert('A referência é: 2I5.')\"&gt;Referência&lt;/span&gt;\n                    \n                    &lt;/div&gt;\n                    \n                    &lt;iframe id=\"fig2-5\" class=\"graficos\" loading=\"lazy\" src=\"interativo/fig2-5.html\"&gt;&lt;/iframe&gt;\n                    &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P29\" title=\"2P29\"&gt;\n            Note que esta definição, em termos de área, é escolhida pois o deslizamento hiperbólico não altera área de figuras\n            no plano (Proposição &lt;a href=\"#nochangearea\"&gt;2.2&lt;/a&gt;) e desta forma um ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; não será alterado quando aplicarmos o\n            deslizamento hiperbólico. Na figura &lt;a href=\"#figanghipdes\"&gt;2.6&lt;/a&gt;, o ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img725.svg\" alt=\"$OAV$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual ao ângulo hiperbólico\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img726.svg\" alt=\"$OA'V'$\" loading=\"lazy\"&gt;&lt;/span&gt; se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img713.svg\" alt=\"$V'$\" loading=\"lazy\"&gt;&lt;/span&gt; são imagens respectivas dos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img310.svg\" alt=\"$V$\" loading=\"lazy\"&gt;&lt;/span&gt; por deslizamento hiperbólico.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figanghipdes\"&gt;&lt;/a&gt;&lt;a id=\"1974\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I6\" title=\"2I6\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.6:&lt;/strong&gt;\n            Ângulo hiperbólico deslizado.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipdes.png\" alt=\"Image anghipdes\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P30\" title=\"2P30\"&gt;\n            Note ainda que podemos considerar ângulos hiperbólicos de qualquer magnitude, já que a área do setor entre as\n            assíntotas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img706.svg\" alt=\"$y = \\pm x$\" loading=\"lazy\"&gt;&lt;/span&gt; e a hipérbole, é infinita. Vamos agora definir seno e cosseno hiperbólico de um ângulo\n            (hiperbólico) &lt;!-- MATH\n             $u \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"2P31\" title=\"2P31\"&gt;\n            Nesses termos, dado um ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, determinado pelo arco de hipérbole &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt;, consideramos o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n            projeção do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img6.svg\" alt=\"$Q$\" loading=\"lazy\"&gt;&lt;/span&gt; projeção do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;. O cosseno hiperbólico de\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;a name=\"1977\"&gt;&lt;/a&gt; é definido como sendo a abscissa do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, o comprimento do segmento\n            orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img74.svg\" alt=\"$OP$\" loading=\"lazy\"&gt;&lt;/span&gt; (ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img75.svg\" alt=\"$QA$\" loading=\"lazy\"&gt;&lt;/span&gt;), com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Note que este segmento orientado nunca terá sentido contrário ao\n            eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, a medida de cosseno hiperbólico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; será sempre positiva (maior ou igual a 1 para ser mais\n            preciso). O seno hiperbólico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;a name=\"1978\"&gt;&lt;/a&gt; é definido como sendo a ordenada do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, o\n            comprimento do segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img73.svg\" alt=\"$PA$\" loading=\"lazy\"&gt;&lt;/span&gt; (ou &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img72.svg\" alt=\"$OQ$\" loading=\"lazy\"&gt;&lt;/span&gt;), com relação ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, se o segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img73.svg\" alt=\"$PA$\" loading=\"lazy\"&gt;&lt;/span&gt; tem\n            sentido contrário ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt;, então entendemos que a medida do segmento é negativa. Isto ocorrerá apenas para valores\n            negativos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n           &lt;a id=\"figsencoship\"&gt;&lt;/a&gt;&lt;a id=\"1981\"&gt;&lt;/a&gt;\n           &lt;div id=\"2I7\" title=\"2I7\" class=\"interativo unidade\"&gt;\n            &lt;div class=\"controles_interatividade\"&gt;\n            \n            &lt;a href=\"interativo/fig2-7.html\" target=\"_blank\" class=\"btn_abrirInterativo\"&gt;Ver maior&lt;/a&gt;&lt;span class=\"barra\"&gt; | &lt;/span&gt;&lt;span class=\"referencia\" onclick=\"alert('A referência é: 2I7.')\"&gt;Referência&lt;/span&gt;\n            \n            &lt;/div&gt;\n            \n            &lt;iframe id=\"fig2-7\" class=\"graficos\" loading=\"lazy\" src=\"interativo/fig2-7.html\"&gt;&lt;/iframe&gt;\n            &lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"2P32\" title=\"2P32\"&gt;\n            Representamos isto escrevendo\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}u = PA \\qquad \\text{e} \\qquad \\cosh u = OP.\n            \\end{displaymath}\n             --&gt;\n           \n            &lt;div class=\"mathdisplay unidade\" id=\"2P33\" title=\"2P33\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img727.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u = PA$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img728.svg\" alt=\"$\\displaystyle \\qquad \\cosh u = OP. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P34\" title=\"2P34\"&gt;\n            As demais funções trigonométricas hiperbólicas, tangente, cotangente, secante e cossecante, são definidas como na\n            trigonometria circular, isto é, respectivamente &lt;a name=\"1985\"&gt;&lt;/a&gt;&lt;a name=\"1986\"&gt;&lt;/a&gt;&lt;a name=\"1987\"&gt;&lt;/a&gt;&lt;a name=\"1988\"&gt;&lt;/a&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P35\" title=\"2P35\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img729.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}u = \\frac{{\\mathrm{senh}}u}{\\cosh u}, \\qquad$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&nbsp;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img730.svg\" alt=\"$\\displaystyle \\qquad {\\mathrm{ctgh}}u = \\frac{\\cosh u}{{\\mathrm{senh}}u},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img731.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}u = \\frac{1}{\\cosh u} \\qquad$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;e&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img732.svg\" alt=\"$\\displaystyle \\qquad {\\mathrm{csch}}u = \\frac{1}{{\\mathrm{senh}}u}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"2P36\" title=\"2P36\"&gt;\n            Se considerarmos dois ângulos hiperbólicos de medidas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img78.svg\" alt=\"$-u$\" loading=\"lazy\"&gt;&lt;/span&gt;, representados respectivamente pelos arcos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img36.svg\" alt=\"$VA$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img733.svg\" alt=\"$VC$\" loading=\"lazy\"&gt;&lt;/span&gt;, vemos (na figura &lt;a href=\"#figangcont\"&gt;2.8&lt;/a&gt;) que os valores de seno hiperbólico são diferentes apenas por um sinal, pois os\n            segmentos orientados &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img72.svg\" alt=\"$OQ$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img734.svg\" alt=\"$OQ'$\" loading=\"lazy\"&gt;&lt;/span&gt; tem sentidos opostos e que os valores de cosseno hiperbólico são ambos iguais ao\n            segmento orientado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img74.svg\" alt=\"$OP$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figangcont\"&gt;&lt;/a&gt;&lt;a id=\"2004\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I8\" title=\"2I8\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.8:&lt;/strong&gt;\n            Ângulos hiperbólicos de sinais contrários.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/sencoshpn.png\" alt=\"Image sencoshpn\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P37\" title=\"2P37\"&gt;\n            Isto significa que,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P38\" title=\"2P38\"&gt;&lt;a id=\"sencoshparimpar\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            {\\mathrm{senh}}u = -{\\mathrm{senh}}(-u), \\qquad \\text{e} \\qquad \\cosh u = \\cosh(-u).\n            \\end{equation}\n             --&gt;\n            &lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img735.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u = -{\\mathrm{senh}}(-u),$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img736.svg\" alt=\"$\\displaystyle \\qquad \\cosh u = \\cosh(-u).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p style=\"text-indent:0px !important;\" class=\" unidade\" id=\"2P39\" title=\"2P39\"&gt;\n            Em outras palavras, o seno hiperbólico é uma função ímpar e o cosseno hiperbólico é uma função par.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"2P40\" title=\"2P40\"&gt;\n            Nosso próximo passo é deduzir as principais fórmulas da trigonometria hiperbólica. Serão cinco fórmulas, contando com\n            as duas identidades em (&lt;a href=\"#sencoshparimpar\"&gt;2.2&lt;/a&gt;). Faltam a relação fundamental e as fórmulas de soma de arcos para o seno\n            e o cosseno hiperbólicos. Demais fórmulas trigonométricas que se deseje podem ser deduzidas a partir destas cinco.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"2P41\" title=\"2P41\"&gt;\n            Da figura &lt;a href=\"#figsencoship\"&gt;2.7&lt;/a&gt;, podemos ver claramente que as coordenadas cartesianas do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; são &lt;!-- MATH\n             $A = (\\cosh u,\n            {\\mathrm{senh}}u)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img737.svg\" alt=\"$A = (\\cosh u,\n            {\\mathrm{senh}}u)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; está sobre a hipérbole, e então suas coordenadas devem obrigatoriamente satisfazer a\n            equação da hipérbole &lt;!-- MATH\n             $x^{2} - y^{2} = 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img704.svg\" alt=\"$x^{2} - y^{2} = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; e, assim,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P42\" title=\"2P42\"&gt;&lt;a id=\"idfundhip\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            \\cosh^{2} u - {\\mathrm{senh}}^{2} u = (\\cosh u)^{2} - ({\\mathrm{senh}}u)^{2} = 1,\n            \\end{equation}\n             --&gt;\n            &lt;table &gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img738.svg\" alt=\"$\\displaystyle \\cosh^{2} u - {\\mathrm{senh}}^{2} u = (\\cosh u)^{2} - ({\\mathrm{senh}}u)^{2} = 1,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p style=\"text-indent:0px !important;\" class=\" unidade\" id=\"2P43\" title=\"2P43\"&gt;\n            que é a relação fundamental da trigonometria hiperbólica.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"2P44\" title=\"2P44\"&gt;\n            Vamos agora mostrar a validade das fórmulas trigonométricas da soma de arcos do cosseno hiperbólico e do seno\n            hiperbólico. Consideremos dois ângulos hiperbólicos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; determinados pelos arcos hiperbólicos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img739.svg\" alt=\"$VZ$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img54.svg\" alt=\"$VB$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            respectivamente, conforme a figura abaixo.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figanghipuv\"&gt;&lt;/a&gt;&lt;a id=\"2024\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I9\" title=\"2I9\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.9:&lt;/strong&gt;\n            Ângulos hiperbólicos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipuv.png\" alt=\"Image anghipuv\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P45\" title=\"2P45\"&gt;\n            Tomando o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, sobre a hipérbole, simétrico do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img740.svg\" alt=\"$Z$\" loading=\"lazy\"&gt;&lt;/span&gt; pelo eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img741.svg\" alt=\"$AZ$\" loading=\"lazy\"&gt;&lt;/span&gt; conjugado ao\n            segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, o prolongamento do segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt; corta o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img741.svg\" alt=\"$AZ$\" loading=\"lazy\"&gt;&lt;/span&gt; em seu ponto médio. Chamemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img742.svg\" alt=\"$W$\" loading=\"lazy\"&gt;&lt;/span&gt;, este\n            ponto médio. &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img742.svg\" alt=\"$W$\" loading=\"lazy\"&gt;&lt;/span&gt; também é a projeção de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img740.svg\" alt=\"$Z$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Considerando as retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt; paralelas as assíntotas\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img706.svg\" alt=\"$y = \\pm x$\" loading=\"lazy\"&gt;&lt;/span&gt;, que passam pelos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img740.svg\" alt=\"$Z$\" loading=\"lazy\"&gt;&lt;/span&gt; respectivamente, temos que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt; se encontram sobre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; no\n            ponto que denotaremos por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"2029\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I10\" title=\"2I10\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.10:&lt;/strong&gt;\n             Segmento conjugado ao ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipuv2.png\" alt=\"Image anghipuv2\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P46\" title=\"2P46\"&gt;\n            Decorre disto que\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\cosh u = OW \\qquad \\text{e} \\qquad {\\mathrm{senh}}u = WZ.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P47\" title=\"2P47\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img743.svg\" alt=\"$\\displaystyle \\cosh u = OW$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img744.svg\" alt=\"$\\displaystyle \\qquad {\\mathrm{senh}}u = WZ. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P48\" title=\"2P48\"&gt;\n            Vamos agora aplicar um deslizamento hiperbólico que desliza o arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img739.svg\" alt=\"$VZ$\" loading=\"lazy\"&gt;&lt;/span&gt; de forma que a imagem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img713.svg\" alt=\"$V'$\" loading=\"lazy\"&gt;&lt;/span&gt; de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img310.svg\" alt=\"$V$\" loading=\"lazy\"&gt;&lt;/span&gt; coincide com\n            o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt;. A imagem de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img740.svg\" alt=\"$Z$\" loading=\"lazy\"&gt;&lt;/span&gt; então será denotada por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img745.svg\" alt=\"$Z' = C$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembrando ainda que o ângulo hiperbólico\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img746.svg\" alt=\"$BOC$\" loading=\"lazy\"&gt;&lt;/span&gt; continua sendo o ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, em virtude da invariância de áreas por deslizamento hiperbólico. O ponto\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt; por sua vez determina o arco de hipérbole &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img733.svg\" alt=\"$VC$\" loading=\"lazy\"&gt;&lt;/span&gt; associado ao ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img135.svg\" alt=\"$(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também, sejam &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img79.svg\" alt=\"$A'$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img747.svg\" alt=\"$W'$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img714.svg\" alt=\"$M'$\" loading=\"lazy\"&gt;&lt;/span&gt; as respectivas imagens dos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img742.svg\" alt=\"$W$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img101.svg\" alt=\"$M$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img715.svg\" alt=\"$r'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img716.svg\" alt=\"$s'$\" loading=\"lazy\"&gt;&lt;/span&gt; as respectivas imagens das retas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figanghipu\"&gt;&lt;/a&gt;&lt;a id=\"2034\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I11\" title=\"2I11\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.11:&lt;/strong&gt;\n             Ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img3.svg\" alt=\"$u+v$\" loading=\"lazy\"&gt;&lt;/span&gt;. &lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipuv3.png\" alt=\"Image anghipuv3\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P49\" title=\"2P49\"&gt;\n            Pelos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt; traçamos, as perpendiculares ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img139.svg\" alt=\"$BP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img748.svg\" alt=\"$CR$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembremos que a corda &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img749.svg\" alt=\"$CA'$\" loading=\"lazy\"&gt;&lt;/span&gt;, é ainda\n            conjugada a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img750.svg\" alt=\"$OB$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou seja, o prolongamento de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img750.svg\" alt=\"$OB$\" loading=\"lazy\"&gt;&lt;/span&gt; encontra o ponto médio do segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img749.svg\" alt=\"$CA'$\" loading=\"lazy\"&gt;&lt;/span&gt; e com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img747.svg\" alt=\"$W'$\" loading=\"lazy\"&gt;&lt;/span&gt; sendo este\n            ponto médio.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"2039\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I12\" title=\"2I12\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.12:&lt;/strong&gt;\n             Projeções &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img5.svg\" alt=\"$R$\" loading=\"lazy\"&gt;&lt;/span&gt;. &lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipuv4.png\" alt=\"Image anghipuv4\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P50\" title=\"2P50\"&gt;\n            Nestes termos temos as relações,\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}v = PB \\qquad \\text{e} \\qquad \\cosh v = OP,\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P51\" title=\"2P51\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img751.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}v = PB$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img752.svg\" alt=\"$\\displaystyle \\qquad \\cosh v = OP, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P52\" style=\"text-indent: 0 !important;\" title=\"2P52\"&gt;\n            e\n        &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}(u+v) = RC \\qquad \\text{e} \\qquad \\cosh(u+v) = OR.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P53\" title=\"2P53\"&gt;\n            &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img753.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}(u+v) = RC$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img754.svg\" alt=\"$\\displaystyle \\qquad \\cosh(u+v) = OR. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P54\" title=\"2P54\"&gt;\n            Vamos agora mostrar que também valem,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P55\" title=\"2P55\"&gt;&lt;a id=\"idsencosh\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            {\\mathrm{senh}}u = \\frac{W'C}{OB} \\qquad \\text{e} \\qquad \\cosh u = \\frac{OW'}{OB}.\n            \\end{equation}\n             --&gt;\n            &lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img755.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u = \\frac{W'C}{OB}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img756.svg\" alt=\"$\\displaystyle \\qquad \\cosh u = \\frac{OW'}{OB}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"2P56\" title=\"2P56\"&gt;\n            Os segmentos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img757.svg\" alt=\"$OW$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt; estão sobre a mesma reta e então do item &lt;!-- MATH\n             $\\mathbf{(c)}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img647.svg\" alt=\"$\\mathbf{(c)}$\" loading=\"lazy\"&gt;&lt;/span&gt; da proposição &lt;a href=\"#prophip\"&gt;2.1&lt;/a&gt; temos que\n            a razão &lt;!-- MATH\n             $\\frac{OW}{OV}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img758.svg\" alt=\"$\\frac{OW}{OV}$\" loading=\"lazy\"&gt;&lt;/span&gt; é preservada pelo deslizamento hiperbólico, ou seja, &lt;!-- MATH\n             $\\frac{OW}{OV} = \\frac{OW'}{OB}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.93ex; vertical-align: -0.83ex; \" src=\"img/img759.svg\" alt=\"$\\frac{OW}{OV} = \\frac{OW'}{OB}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Levando\n            em conta que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img760.svg\" alt=\"$OV = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos imediatamente que\n            &lt;/p&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\cosh u = OW = \\frac{OW}{OV} = \\frac{OW'}{OB}.\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P57\" title=\"2P57\"&gt;\n            &lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img761.svg\" alt=\"$\\displaystyle \\cosh u = OW = \\frac{OW}{OV} = \\frac{OW'}{OB}. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P58\" title=\"2P58\"&gt;\n            Agora, os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img762.svg\" alt=\"$AMZ$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img763.svg\" alt=\"$A'M'C$\" loading=\"lazy\"&gt;&lt;/span&gt; são triângulos retângulos e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img742.svg\" alt=\"$W$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img747.svg\" alt=\"$W'$\" loading=\"lazy\"&gt;&lt;/span&gt; são pontos médios das respectivas\n            hipotenusas. O ponto médio da hipotenusa é equidistante aos vértices de um triângulo retângulo, isto é, &lt;!-- MATH\n             $WA = WM = WZ$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img764.svg\" alt=\"$WA = WM = WZ$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            e &lt;!-- MATH\n             $W'A' = W'M' = W'C$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img765.svg\" alt=\"$W'A' = W'M' = W'C$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também, como os segmentos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img766.svg\" alt=\"$MW$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt; estão sobre uma mesma reta, a razão &lt;!-- MATH\n             $\\frac{MW}{OV}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img767.svg\" alt=\"$\\frac{MW}{OV}$\" loading=\"lazy\"&gt;&lt;/span&gt; é\n            preservada pelo deslizamento hiperbólico, isto é, &lt;!-- MATH\n             $\\frac{MW}{OV} = \\frac{M'W'}{OV'}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.93ex; vertical-align: -0.83ex; \" src=\"img/img768.svg\" alt=\"$\\frac{MW}{OV} = \\frac{M'W'}{OV'}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Segue que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}u = WZ = \\frac{WZ}{OV} = \\frac{MW}{OV} = \\frac{M'W'}{OB} = \\frac{W'C}{OB},\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P59\" title=\"2P59\"&gt;\n            &lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img769.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u = WZ = \\frac{WZ}{OV} = \\frac{MW}{OV} = \\frac{M'W'}{OB} = \\frac{W'C}{OB}, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            e isto garante as igualdades (&lt;a href=\"#idsencosh\"&gt;2.4&lt;/a&gt;).\n            \n            &lt;p class=\" unidade\" id=\"2P60\" title=\"2P60\"&gt;\n            Seja &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img6.svg\" alt=\"$Q$\" loading=\"lazy\"&gt;&lt;/span&gt; a projeção de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img747.svg\" alt=\"$W'$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img7.svg\" alt=\"$S$\" loading=\"lazy\"&gt;&lt;/span&gt; a projeção de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img747.svg\" alt=\"$W'$\" loading=\"lazy\"&gt;&lt;/span&gt; sobre o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img770.svg\" alt=\"$RC$\" loading=\"lazy\"&gt;&lt;/span&gt;, conforme a figura\n            &lt;a href=\"#figanghipuv2\"&gt;2.13&lt;/a&gt;.\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figanghipuv2\"&gt;&lt;/a&gt;&lt;a id=\"2081\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I13\" title=\"2I13\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.13:&lt;/strong&gt;\n             Projeções &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img6.svg\" alt=\"$Q$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img7.svg\" alt=\"$S$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipuv5.png\" alt=\"Image anghipuv5\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P61\" title=\"2P61\"&gt;\n            Notemos que os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img771.svg\" alt=\"$OPB$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img772.svg\" alt=\"$OQW'$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes. Vamos verificar que também são semelhantes os triângulos\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img771.svg\" alt=\"$OPB$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img773.svg\" alt=\"$CSW'$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para isto mostraremos que o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img774.svg\" alt=\"$B\\hat{O}P$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual ao ângulo &lt;!-- MATH\n             $S\\hat{C}W'$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img775.svg\" alt=\"$S\\hat{C}W'$\" loading=\"lazy\"&gt;&lt;/span&gt;. A reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img716.svg\" alt=\"$s'$\" loading=\"lazy\"&gt;&lt;/span&gt;, paralela a\n            bissetriz &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img708.svg\" alt=\"$y=x$\" loading=\"lazy\"&gt;&lt;/span&gt;, passa por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\" loading=\"lazy\"&gt;&lt;/span&gt;, intercepta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img750.svg\" alt=\"$OB$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img714.svg\" alt=\"$M'$\" loading=\"lazy\"&gt;&lt;/span&gt; e intercepta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img132.svg\" alt=\"$OV$\" loading=\"lazy\"&gt;&lt;/span&gt; em um ponto que chamaremos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img8.svg\" alt=\"$N$\" loading=\"lazy\"&gt;&lt;/span&gt; (Figura\n            &lt;a href=\"#figanghipuv3\"&gt;2.14&lt;/a&gt;).\n            &lt;/p&gt;\n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figanghipuv3\"&gt;&lt;/a&gt;&lt;a id=\"2089\"&gt;&lt;/a&gt;\n            &lt;table id=\"2I14\" title=\"2I14\"&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.14:&lt;/strong&gt;\n            Projeção &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img8.svg\" alt=\"$N$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/anghipuv6.png\" alt=\"Image anghipuv6\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P62\" title=\"2P62\"&gt;\n            Assim, &lt;!-- MATH\n             $R\\hat{C}N = R\\hat{N}C = 45^{\\circ}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.29ex; vertical-align: -0.13ex; \" src=\"img/img776.svg\" alt=\"$R\\hat{C}N = R\\hat{N}C = 45^{\\circ}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mais ainda, como o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img777.svg\" alt=\"$CM'A'$\" loading=\"lazy\"&gt;&lt;/span&gt; é retângulo em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img714.svg\" alt=\"$M'$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img747.svg\" alt=\"$W'$\" loading=\"lazy\"&gt;&lt;/span&gt; é o ponto\n            médio da hipotenusa &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img749.svg\" alt=\"$CA'$\" loading=\"lazy\"&gt;&lt;/span&gt;, segue que o triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img778.svg\" alt=\"$M'W'C$\" loading=\"lazy\"&gt;&lt;/span&gt; é isósceles e pontanto os ângulos &lt;!-- MATH\n             $W'\\hat{M'}C$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.33ex; vertical-align: -0.10ex; \" src=\"img/img779.svg\" alt=\"$W'\\hat{M'}C$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n            &lt;!-- MATH\n             $W'\\hat{C}M'$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img780.svg\" alt=\"$W'\\hat{C}M'$\" loading=\"lazy\"&gt;&lt;/span&gt; possuem a mesma medida, isto é, &lt;!-- MATH\n             $W'\\hat{M'}C = W'\\hat{C}M'$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.33ex; vertical-align: -0.10ex; \" src=\"img/img781.svg\" alt=\"$W'\\hat{M'}C = W'\\hat{C}M'$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mas &lt;!-- MATH\n             \\begin{displaymath}\n            M'\\hat{N}P + M'\\hat{N}O = 180^{\\circ}\n            = M'\\hat{N}O + M'\\hat{O}N + N\\hat{M'}O, \n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P63\" title=\"2P63\"&gt;\n            &lt;img style=\"height: 2.50ex; vertical-align: -0.27ex; \" src=\"img/img782.svg\" alt=\"$\\displaystyle M'\\hat{N}P + M'\\hat{N}O = 180^{\\circ}\n            = M'\\hat{N}O + M'\\hat{O}N + N\\hat{M'}O, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P64\" style=\"text-indent: 0;\" title=\"2P64\"&gt;\n            e, portanto, &lt;!-- MATH\n             $M'\\hat{N}P = N\\hat{M'}O + M'\\hat{O}N = N\\hat{M'}O + M'\\hat{O}P$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.27ex; \" src=\"img/img783.svg\" alt=\"$M'\\hat{N}P = N\\hat{M'}O + M'\\hat{O}N = N\\hat{M'}O + M'\\hat{O}P$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            Segue disto que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            B\\hat{O}P = M'\\hat{O}P = M'\\hat{N}P - N\\hat{M'}O = C\\hat{N}R - C\\hat{M'}W'.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P65\" title=\"2P65\"&gt;\n            &lt;img style=\"height: 2.50ex; vertical-align: -0.27ex; \" src=\"img/img784.svg\" alt=\"$\\displaystyle B\\hat{O}P = M'\\hat{O}P = M'\\hat{N}P - N\\hat{M'}O = C\\hat{N}R - C\\hat{M'}W'. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P66\" style=\"text-indent: 0 !important;\" title=\"2P66\"&gt;\n            Agora, &lt;!-- MATH\n             $S\\hat{C}W' = R\\hat{C}N - M'\\hat{C}W'$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.27ex; \" src=\"img/img785.svg\" alt=\"$S\\hat{C}W' = R\\hat{C}N - M'\\hat{C}W'$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            B\\hat{O}P = C\\hat{N}R - C\\hat{M'}W' = C\\hat{N}R - M'\\hat{C}W' = C\\hat{N}R + S\\hat{C}W' - R\\hat{C}N,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P67\" title=\"2P67\"&gt;\n            &lt;img style=\"height: 2.50ex; vertical-align: -0.27ex; \" src=\"img/img786.svg\" alt=\"$\\displaystyle B\\hat{O}P = C\\hat{N}R - C\\hat{M'}W' = C\\hat{N}R - M'\\hat{C}W' = C\\hat{N}R + S\\hat{C}W' - R\\hat{C}N, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P68\" style=\"text-indent: 0 !important;\" title=\"2P68\"&gt;\n            e como &lt;!-- MATH\n             $C\\hat{N}R = R\\hat{C}N$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img787.svg\" alt=\"$C\\hat{N}R = R\\hat{C}N$\" loading=\"lazy\"&gt;&lt;/span&gt; então temos,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            B\\hat{O}P = S\\hat{C}W',\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P69\" title=\"2P69\"&gt;\n            &lt;img style=\"height: 2.26ex; vertical-align: -0.10ex; \" src=\"img/img788.svg\" alt=\"$\\displaystyle B\\hat{O}P = S\\hat{C}W', $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P70\" style=\"text-indent: 0 !important;\" title=\"2P70\"&gt;\n            como desejado. Isto mostra que os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img789.svg\" alt=\"$BOP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img773.svg\" alt=\"$CSW'$\" loading=\"lazy\"&gt;&lt;/span&gt; são semelhantes. Desta semelhança, segue que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\frac{SC}{CW'} = \\frac{OP}{OB} \\qquad \\text{e} \\qquad \\frac{W'S}{W'C} = \\frac{PB}{OB}\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P71\" title=\"2P71\"&gt;\n            &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img790.svg\" alt=\"$\\displaystyle \\frac{SC}{CW'} = \\frac{OP}{OB}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img791.svg\" alt=\"$\\displaystyle \\qquad \\frac{W'S}{W'C} = \\frac{PB}{OB} $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P72\" style=\"text-indent: 0 !important;\" title=\"2P72\"&gt;\n            e destas igualdades,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            SC = \\frac{CW'}{OB} OP \\qquad \\text{e} \\qquad W'S = \\frac{W'C}{OB} PB.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P73\" title=\"2P73\"&gt;\n            &lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img792.svg\" alt=\"$\\displaystyle SC = \\frac{CW'}{OB} OP$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img793.svg\" alt=\"$\\displaystyle \\qquad W'S = \\frac{W'C}{OB} PB. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P74\" title=\"2P74\"&gt;\n            Também são semelhantes os triângulos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img789.svg\" alt=\"$BOP$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img794.svg\" alt=\"$W'OQ$\" loading=\"lazy\"&gt;&lt;/span&gt; e desta semelhança, temos\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\frac{QW'}{OW'} = \\frac{PB}{OB} \\qquad \\text{e} \\qquad \\frac{OQ}{OW'} = \\frac{OP}{OB}\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P75\" title=\"2P75\"&gt;\n            &lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img795.svg\" alt=\"$\\displaystyle \\frac{QW'}{OW'} = \\frac{PB}{OB}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img796.svg\" alt=\"$\\displaystyle \\qquad \\frac{OQ}{OW'} = \\frac{OP}{OB} $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P76\" style=\"text-indent: 0 !important;\" title=\"2P76\"&gt;\n            e destas igualdades,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            QW' = \\frac{OW'}{OB} PB \\qquad \\text{e} \\qquad OQ = \\frac{OW'}{OB} OP.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P77\" title=\"2P77\"&gt;\n            &lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img797.svg\" alt=\"$\\displaystyle QW' = \\frac{OW'}{OB} PB$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img798.svg\" alt=\"$\\displaystyle \\qquad OQ = \\frac{OW'}{OB} OP. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"2P78\" title=\"2P78\"&gt;\n            Finalmente, lembrando que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}v = PB, \\qquad \\cosh v = OP,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P79\" title=\"2P79\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img799.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}v = PB, \\qquad \\cosh v = OP, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}u = \\frac{W'C}{OB}, \\qquad \\cosh u = \\frac{OW'}{OB},\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P80\" title=\"2P80\"&gt;\n            &lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img800.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u = \\frac{W'C}{OB}, \\qquad \\cosh u = \\frac{OW'}{OB}, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}(u+v) = RC \\qquad \\text{e} \\qquad \\cosh(u+v) = OR,\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P81\" title=\"2P81\"&gt;\n            &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img753.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}(u+v) = RC$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img801.svg\" alt=\"$\\displaystyle \\qquad \\cosh(u+v) = OR, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P82\" style=\"text-indent: 0 !important;\" title=\"2P82\"&gt;\n            temos,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P83\" title=\"2P83\"&gt;&lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;width: 39%\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img802.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.50ex; \" src=\"img/img803.svg\" alt=\"$\\displaystyle = RC = QW' + SC$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img804.svg\" alt=\"$\\displaystyle = \\frac{CW'}{OB} OP + \\frac{OW'}{OB} PB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img805.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}u \\cosh v + \\cosh u {\\mathrm{senh}}v,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &lt;a id=\"addarcsenh\"&gt;(&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"2P84\" style=\"text-indent: 0 !important;\" title=\"2P84\"&gt;\n            e também,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P85\" title=\"2P85\"&gt;&lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right; width: 39%;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img806.svg\" alt=\"$\\displaystyle \\cosh(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.50ex; \" src=\"img/img807.svg\" alt=\"$\\displaystyle = OR = OQ + W'S$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.68ex; vertical-align: -1.55ex; \" src=\"img/img808.svg\" alt=\"$\\displaystyle = \\frac{OW'}{OB} OP + \\frac{W'C}{OB} PB$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img809.svg\" alt=\"$\\displaystyle = \\cosh u \\cosh v + {\\mathrm{senh}}u {\\mathrm{senh}}v.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &lt;a id=\"addarccosh\"&gt;(&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"2P86\" title=\"2P86\"&gt;\n            As fórmulas (&lt;a href=\"#addarcsenh\"&gt;2.5&lt;/a&gt;) e (&lt;a href=\"#addarccosh\"&gt;2.6&lt;/a&gt;), juntamente com a relação fundamental (&lt;a href=\"#idfundhip\"&gt;2.3&lt;/a&gt;) e as duas\n            fórmulas em (&lt;a href=\"#sencoshparimpar\"&gt;2.2&lt;/a&gt;), constituem as 5 fórmulas básicas da trigonometria hiperbólica. Com elas podemos\n            deduzir outras fórmulas, como por exemplo, as fórmulas de duplicação de arcos,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P87\" title=\"2P87\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img810.svg\" alt=\"$\\displaystyle \\cosh(2u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img811.svg\" alt=\"$\\displaystyle = \\cosh(u+u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img812.svg\" alt=\"$\\displaystyle = \\cosh^{2} u + {\\mathrm{senh}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n           \n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img813.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}(2u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img814.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}(u+u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img815.svg\" alt=\"$\\displaystyle = 2{\\mathrm{senh}}u \\cosh u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"2P88\" style=\"text-indent: 0 !important;\" title=\"2P88\"&gt;\n            e as fórmulas de diferença de arcos,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P89\" title=\"2P89\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img816.svg\" alt=\"$\\displaystyle \\cosh(u - v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img817.svg\" alt=\"$\\displaystyle = \\cosh u \\cosh(-v) + {\\mathrm{senh}}u {\\mathrm{senh}}(-v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img818.svg\" alt=\"$\\displaystyle = \\cosh u \\cosh v - {\\mathrm{senh}}u {\\mathrm{senh}}v$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img819.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}(u - v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img820.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}u \\cosh(-v) + {\\mathrm{senh}}(-v) \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img821.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}u \\cosh v - {\\mathrm{senh}}v \\cosh u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"2P90\" title=\"2P90\"&gt;\n            Vamos agora obter duas outras fórmulas trigonométricas hiperbólicas que serão úteis mais adiante. São fórmulas fáceis\n            de serem obtidas, similares às fórmulas obtidas na proposição &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#propidsectg\"&gt;1.1&lt;/a&gt;. Estamos apresentando-as em virtude do\n            uso futuro (na seção &lt;a href=\"#secderhipinv\"&gt;2.8&lt;/a&gt;).\n            &lt;/p&gt;\n            &lt;div id=\"2Teo3\" title=\"2Teo3\" class=\" unidade\"&gt;&lt;a id=\"propidsectghip\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;..&lt;span class=\"arabic\"&gt;3&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n            &lt;i&gt;São válidas as seguintes identidades trigonométricas hiperbólicas\n            &lt;/i&gt;&lt;table width=\"90%\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img184.svg\" alt=\"$(i)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Para todos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; reais,\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P91\" title=\"2P91\"&gt;&lt;a id=\"idtghadd\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            {\\mathrm {tgh}}(u+v) = \\frac{{\\mathrm {tgh}}u + {\\mathrm {tgh}}v}{1 + {\\mathrm {tgh}}u {\\mathrm {tgh}}v}.\n            \\end{equation}\n             --&gt;\n            &lt;table &gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img822.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}(u+v) = \\frac{{\\mathrm {tgh}}u + {\\mathrm {tgh}}v}{1 + {\\mathrm {tgh}}u {\\mathrm {tgh}}v}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            &lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img187.svg\" alt=\"$(ii)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Para todo &lt;!-- MATH\n             $u \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P92\" title=\"2P92\"&gt;&lt;a id=\"idtghsech\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            1 - {\\mathrm {tgh}}^{2} u = {\\mathrm{sech}}^{2} u.\n            \\end{equation}\n             --&gt;\n            &lt;table &gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img823.svg\" alt=\"$\\displaystyle 1 - {\\mathrm {tgh}}^{2} u = {\\mathrm{sech}}^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            &lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img190.svg\" alt=\"$(iii)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Para todo &lt;!-- MATH\n             $u \\in \\mathbb{R}-\\{0\\}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img824.svg\" alt=\"$u \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P93\" title=\"2P93\"&gt;&lt;a id=\"idctghcsch\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            {\\mathrm{ctgh}}^{2} u - 1 = {\\mathrm{csch}}^{2} u\n            \\end{equation}\n             --&gt;\n            &lt;table &gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img825.svg\" alt=\"$\\displaystyle {\\mathrm{ctgh}}^{2} u - 1 = {\\mathrm{csch}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            (&lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;9&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n            \n            Para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img184.svg\" alt=\"$(i)$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"2P94\" title=\"2P94\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img826.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}(u+v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img827.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm{senh}}(u+v)}{\\cosh(u+v)} = \\frac{{\\mathrm{senh}}...\n            ... {\\mathrm{senh}}v \\cosh u}{\\cosh u \\cosh v + {\\mathrm{senh}}u {\\mathrm{senh}}v}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.48ex; vertical-align: -2.52ex; \" src=\"img/img828.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm{senh}}u \\cosh v + {\\mathrm{senh}}v \\cosh u}{\\cosh u \\cosh v (1 + \\frac{{\\mathrm{senh}}u {\\mathrm{senh}}v}{\\cosh u \\cosh v}) }$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.65ex; vertical-align: -2.52ex; \" src=\"img/img829.svg\" alt=\"$\\displaystyle = \\left( \\frac{{\\mathrm{senh}}u \\cosh v + {\\mathrm{senh}}v \\cosh ...\n            ...rac{1}{(1 + \\frac{{\\mathrm{senh}}u}{\\cosh u} \\frac{{\\mathrm{senh}}v}{\\cosh v})}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img830.svg\" alt=\"$\\displaystyle = \\left( \\frac{{\\mathrm{senh}}u}{\\cosh u} + \\frac{{\\mathrm{senh}}v}{\\cosh v} \\right) \\frac{1}{(1 + {\\mathrm {tgh}}u {\\mathrm {tgh}}v)}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img831.svg\" alt=\"$\\displaystyle = \\left( {\\mathrm {tgh}}u + {\\mathrm {tgh}}v \\right) \\frac{1}{(1 + {\\mathrm {tgh}}u {\\mathrm {tgh}}v)}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img832.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {tgh}}u + {\\mathrm {tgh}}v}{1 + {\\mathrm {tgh}}u {\\mathrm {tgh}}v}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"2P95\" style=\"text-indent: 0 !important;\" title=\"2P95\"&gt;\n            Os itens &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img187.svg\" alt=\"$(ii)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img190.svg\" alt=\"$(iii)$\" loading=\"lazy\"&gt;&lt;/span&gt; ficam\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"2P96\" title=\"2P96\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img833.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.79ex; vertical-align: -1.82ex; \" src=\"img/img834.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cosh^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -1.82ex; \" src=\"img/img835.svg\" alt=\"$\\displaystyle = \\frac{\\cosh^{2}u - {\\mathrm{senh}}^{2}u}{\\cosh^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -1.82ex; \" src=\"img/img836.svg\" alt=\"$\\displaystyle = 1 - \\frac{{\\mathrm{senh}}^{2}u}{\\cosh^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img837.svg\" alt=\"$\\displaystyle = 1- {\\mathrm {tgh}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;\n            &lt;/table&gt;\n            &lt;p style=\"text-indent: 0 !important; text-align: left !important;\"&gt;&lt;span class=\"MATH\"&gt;e&lt;/span&gt;&lt;/p&gt;\n            &lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img838.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.79ex; vertical-align: -1.82ex; \" src=\"img/img839.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm{senh}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -1.82ex; \" src=\"img/img840.svg\" alt=\"$\\displaystyle = \\frac{\\cosh^{2}u - {\\mathrm{senh}}^{2}u}{{\\mathrm{senh}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -1.82ex; \" src=\"img/img841.svg\" alt=\"$\\displaystyle = \\frac{\\cosh^{2}u}{{\\mathrm{senh}}^{2} u} - 1$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img842.svg\" alt=\"$\\displaystyle = {\\mathrm{ctgh}}^{2} u - 1$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n            &lt;p style=\"text-indent: 0 !important;\"&gt;&lt;span class=\"MATH\"&gt;e a prova está concluída.&lt;/span&gt;&lt;span style=\"text-align: right;\"&gt;                 &lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; float: right;\" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;\n            &lt;/span&gt;&lt;/p&gt;\n           \n           \n            &lt;/div&gt;",
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      "Capítulo 2: Funções trigonométricas hiperbólicas"
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    "href": "funcoes-trigonometricas-hiperbolicas.html#SECTION00630000000000000000",
    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.3 As funções trigonométricas hiperbólicas",
    "text": "2.3 As funções trigonométricas hiperbólicas\n\n          &lt;a id=\"secfunchip\"&gt;&lt;/a&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P97\" title=\"2P97\"&gt;\n                Nesta seção, vamos estudar os aspectos das funções trigonométricas hiperbólicas. Primeiro vamos observar os gráficos dessas funções, determinando, com precisão, os respectivos domínios. Também, vamos observar alguns limites importantes em cada uma das funções. Para esse nosso estudo, vamos considerar as funções de uma variável real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; que a cada valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; associa o seno, ou o cosseno, ou a tangente, ou a cotangente, ou a secante, ou ainda a cossecante hiperbólica de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Vamos olhar uma a uma.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P98\" title=\"2P98\"&gt;\n                Para a função &lt;!-- MATH\n                 $w = f(u) = {\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img843.svg\" alt=\"$w = f(u) = {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, notemos que para cada valor real de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, construímos o ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                determinado pelo arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt;, onde a ordenada do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; é o seno hiperbólico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;a name=\"2263\"&gt;&lt;/a&gt;\n                Não há nenhuma impossibilidade matemática para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, o domínio da função &lt;!-- MATH\n                 $f(u) = {\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img844.svg\" alt=\"$f(u) = {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é todo o conjunto\n                dos números reais. Além disso, fazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; variar no conjunto dos reais, os valores resultantes para a ordenada do\n                ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; também percorrem o conjunto dos números reais. Desta forma, temos que a função\n                &lt;/p&gt;\n                 &lt;br&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P99\" title=\"2P99\"&gt;\n                &lt;!-- MATH\n                 \\begin{eqnarray*}\n                f: \\mathbb{R}& \\to & \\mathbb{R}\\\\\n                u & \\mapsto & w = f(u) = {\\mathrm{senh}}u\n                \\end{eqnarray*}\n                 --&gt;\n                &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n                &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img210.svg\" alt=\"$\\displaystyle f : \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img212.svg\" alt=\"$\\displaystyle \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n                &nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img845.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n                &nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                &lt;br clear=\"ALL\"&gt;\n                \n                é sobrejetiva. Além disso, para cada &lt;!-- MATH\n                 $w \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img457.svg\" alt=\"$w \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, é único o valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; que satisfaz &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img846.svg\" alt=\"$w = f(u)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então esta função é\n                também injetora. Logo, &lt;!-- MATH\n                 ${\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função bijetora.\n                \n                &lt;p class=\" unidade\" id=\"2P100\" title=\"2P100\"&gt;\n                Conforme &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; aumenta (para o infinito) o tamanho do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt; também aumenta. Por conseguinte, a ordenada do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                aumenta e o valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img218.svg\" alt=\"$f(u)$\" loading=\"lazy\"&gt;&lt;/span&gt; também aumenta indefinidamente. O mesmo ocorre para os valores negativos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos\n                então\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to \\infty} {\\mathrm{senh}}u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to -\\infty} {\\mathrm{senh}}u = -\\infty.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P101\" title=\"2P101\"&gt;\n                &lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img848.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\infty} {\\mathrm{senh}}u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.28ex; vertical-align: -1.73ex; \" src=\"img/img849.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to -\\infty} {\\mathrm{senh}}u = -\\infty. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P102\" title=\"2P102\"&gt;\n                O gráfico de &lt;!-- MATH\n                 $f(u) = {\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img844.svg\" alt=\"$f(u) = {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é dado por\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2271\"&gt;&lt;/a&gt;\n                &lt;table id=\"2I15\" title=\"2I15\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.15:&lt;/strong&gt;\n                Gráfico da função seno hiperbólico.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/fsinh.png\" alt=\"Image fsinh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P103\" title=\"2P103\"&gt;\n                Podemos notar ainda que é uma função contínua (mostraremos isto formalmente na próxima seção), ímpar e estritamente\n                crescente.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P104\" title=\"2P104\"&gt;\n                Agora a função &lt;!-- MATH\n                 $w = f(u) = \\cosh u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img850.svg\" alt=\"$w = f(u) = \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;a name=\"2273\"&gt;&lt;/a&gt; Para qualquer valor real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, construímos o arco\n                hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt; associado ao ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, cujo cosseno hiperbólico é a abscissa do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt;. Notemos que\n                não há nenhuma impossibilidade matemática para o valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, sendo assim, o domínio da função &lt;!-- MATH\n                 $f(u) = \\cosh u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img851.svg\" alt=\"$f(u) = \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; é o\n                conjunto dos números reais. Fazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; variar no conjunto dos números reais, vemos que a abscissa do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; somente\n                poderá assumir valores maiores do que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, &lt;!-- MATH\n                 $\\cosh u \\in [1, \\infty)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img852.svg\" alt=\"$\\cosh u \\in [1, \\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto significa que esta função não é\n                sobrejetora no conjunto dos números reais, mas sim no conjunto &lt;!-- MATH\n                 $[1,\\infty)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Note também que esta função não é\n                injetora, pois para qualquer valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; temos &lt;!-- MATH\n                 $\\cosh u = \\cosh(-u)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img853.svg\" alt=\"$\\cosh u = \\cosh(-u)$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é, é uma função par. Portanto a função\n                &lt;/p&gt;\n                 &lt;br&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P105\" title=\"2P105\"&gt;\n                &lt;!-- MATH\n                 \\begin{eqnarray*}\n                f: \\mathbb{R}& \\to & \\mathbb{R}\\\\\n                u & \\mapsto & w = f(u) = \\cosh u\n                \\end{eqnarray*}\n                 --&gt;\n                &lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n                &lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img210.svg\" alt=\"$\\displaystyle f : \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img212.svg\" alt=\"$\\displaystyle \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n                &nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img854.svg\" alt=\"$\\displaystyle w = f(u) = \\cosh u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n                &nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                &lt;br clear=\"ALL\"&gt;\n                &lt;p class=\" unidade\" id=\"2P106\" style=\"text-indent: 0 !important;\" title=\"2P106\"&gt;\n                não é bijetora.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P107\" title=\"2P107\"&gt;\n                Conforme o valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; aumenta (para o infinito), o tamanho do arco &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt; aumenta e a abscissa do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; também\n                aumenta indefinidamente. Uma análise similar para valores negativos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; levam à mesma conclusão. Temos assim,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to \\infty} \\cosh u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to -\\infty} \\cosh u = \\infty.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P108\" title=\"2P108\"&gt;\n                &lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img855.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\infty} \\cosh u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.28ex; vertical-align: -1.73ex; \" src=\"img/img856.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to -\\infty} \\cosh u = \\infty. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P109\" title=\"2P109\"&gt;\n                O gráfico desta função é dado por\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2281\"&gt;&lt;/a&gt;\n                &lt;table id=\"2I16\" title=\"2I16\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.16:&lt;/strong&gt;\n                Gráfico da função cosseno hiperbólico.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/fcosh.png\" alt=\"Image fcosh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P110\" title=\"2P110\"&gt;\n                Para a função &lt;!-- MATH\n                 $w = f(u) = {\\mathrm {tgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img857.svg\" alt=\"$w = f(u) = {\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; vamos usar a identidade &lt;!-- MATH\n                 ${\\mathrm {tgh}}u = \\frac{{\\mathrm{senh}}u}{\\cosh u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img858.svg\" alt=\"${\\mathrm {tgh}}u = \\frac{{\\mathrm{senh}}u}{\\cosh u}$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;a name=\"2285\"&gt;&lt;/a&gt; Fazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; variar no conjunto dos números reais, temos apenas que nos preocupar com o denominador,\n                que não pode ser nulo. Como vimos anteriormente, para qualquer valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; é maior ou igual a 1,\n                e, portanto, o denominador da fração anterior, não se anula. Com isto o domínio da função &lt;!-- MATH\n                 $f(u) = {\\mathrm {tgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img860.svg\" alt=\"$f(u) = {\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é todo o\n                conjunto dos números reais.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P111\" title=\"2P111\"&gt;\n                Além disso, como o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; está entre as retas assíntotas &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img708.svg\" alt=\"$y=x$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img709.svg\" alt=\"$y=-x$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos que a abscissa do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; é\n                sempre maior que a ordenada do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; em módulo, isto é, &lt;!-- MATH\n                 $\\cosh u &gt; |{\\mathrm{senh}}u|$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img861.svg\" alt=\"$\\cosh u &gt; \\vert{\\mathrm{senh}}u\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt; para qualquer valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto\n                significa que a fração &lt;!-- MATH\n                 ${\\mathrm {tgh}}u = \\frac{{\\mathrm{senh}}u}{\\cosh u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img858.svg\" alt=\"${\\mathrm {tgh}}u = \\frac{{\\mathrm{senh}}u}{\\cosh u}$\" loading=\"lazy\"&gt;&lt;/span&gt; resultará sempre valores menores que 1 em módulo, isto é,\n                &lt;!-- MATH\n                 ${\\mathrm {tgh}}u \\in (-1,1)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img862.svg\" alt=\"${\\mathrm {tgh}}u \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P112\" title=\"2P112\"&gt;\n                À medida que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; aumenta indefinidamente, os valores de &lt;!-- MATH\n                 ${\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; tendem a se igualar, pois o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; se\n                aproxima da assíntota &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img708.svg\" alt=\"$y=x$\" loading=\"lazy\"&gt;&lt;/span&gt; e isto significa que quando &lt;!-- MATH\n                 $u \\to \\infty$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img863.svg\" alt=\"$u \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; os valores de &lt;!-- MATH\n                 ${\\mathrm {tgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img864.svg\" alt=\"${\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximam de 1. No\n                caso em que &lt;!-- MATH\n                 $u \\to -\\infty$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img865.svg\" alt=\"$u \\to -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; então o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima da assíntota &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img709.svg\" alt=\"$y=-x$\" loading=\"lazy\"&gt;&lt;/span&gt; e neste caso levamos em conta os sinais\n                de &lt;!-- MATH\n                 ${\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Em outras palavras,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to \\infty} {\\mathrm {tgh}}u = 1 \\qquad \\text{e} \\qquad \\lim_{u \\to -\\infty} {\\mathrm {tgh}}u = -1.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P113\" title=\"2P113\"&gt;\n                &lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img866.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\infty} {\\mathrm {tgh}}u = 1$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.28ex; vertical-align: -1.73ex; \" src=\"img/img867.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to -\\infty} {\\mathrm {tgh}}u = -1. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P114\" title=\"2P114\"&gt;\n                O gráfico da função &lt;!-- MATH\n                 $w = f(u) = {\\mathrm {tgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img857.svg\" alt=\"$w = f(u) = {\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é dado por\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2293\"&gt;&lt;/a&gt;\n                &lt;table id=\"2I17\" title=\"2I17\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.17:&lt;/strong&gt;\n                Gráfico da função tangente hiperbólica.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/ftgh.png\" alt=\"Image ftgh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P115\" title=\"2P115\"&gt;\n                A função &lt;!-- MATH\n                 $f(u) = {\\mathrm {tgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img860.svg\" alt=\"$f(u) = {\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função monótona crescente, ímpar, limitada e bijetora de &lt;!-- MATH\n                 $\\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P116\" title=\"2P116\"&gt;\n                O estudo da função cotangente hiperbólica, &lt;!-- MATH\n                 $w = f(u) = {\\mathrm{ctgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img868.svg\" alt=\"$w = f(u) = {\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, também será feito analizando a identidade &lt;!-- MATH\n                 ${\\mathrm{ctgh}}u =\n                \\frac{\\cosh u}{{\\mathrm{senh}}u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img869.svg\" alt=\"${\\mathrm{ctgh}}u =\n                \\frac{\\cosh u}{{\\mathrm{senh}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;a name=\"2297\"&gt;&lt;/a&gt; Para determinar o domínio desta função, como se trata de\n                um quociente, precisamos nos preocupar com o anulamento do denominador. O seno hiperbólico se anula somente no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, o domínio de &lt;!-- MATH\n                 $f(u) = {\\mathrm{ctgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img870.svg\" alt=\"$f(u) = {\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é o conjunto &lt;!-- MATH\n                 $\\mathbb{R}^{*} = \\mathbb{R}- \\{ 0 \\}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img871.svg\" alt=\"$\\mathbb{R}^{*} = \\mathbb{R}- \\{ 0 \\}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P117\" title=\"2P117\"&gt;\n                Também, como visto anteriormente, o numerador é sempre maior que o denominador, em módulo. Portanto os valores\n                resultantes deste quociente são sempre maiores que 1, em módulo, isto é, a imagem desta função é o conjunto &lt;!-- MATH\n                 $(-\\infty,\n                -1) \\cup (1, \\infty)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. À medida que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce indefinidamente, os valores de &lt;!-- MATH\n                 ${\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproximam (veja\n                explicação anterior) e, portanto,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to \\infty} {\\mathrm{ctgh}}u = 1 \\qquad \\text{e} \\qquad \\lim_{u \\to -\\infty} {\\mathrm{ctgh}}u = -1.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P118\" title=\"2P118\"&gt;\n                &lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img872.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\infty} {\\mathrm{ctgh}}u = 1$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.28ex; vertical-align: -1.73ex; \" src=\"img/img873.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to -\\infty} {\\mathrm{ctgh}}u = -1. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P119\" title=\"2P119\"&gt;\n                Como o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; é um ponto crítico desta função, vamos estudar os limites no ponto 0. A medida que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; se aproxima\n                de &lt;span class=\"MATH\"&gt;0&lt;/span&gt;, os valores do denomidador &lt;!-- MATH\n                 ${\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, se aproximam de &lt;span class=\"MATH\"&gt;0&lt;/span&gt; e a fração vai para o infinito. Temos então, com o\n                respectivo estudo de sinal lateral,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0^{+}} {\\mathrm{ctgh}}u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to 0^{-}} {\\mathrm{ctgh}}u = -\\infty.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P120\" title=\"2P120\"&gt;\n                &lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img874.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} {\\mathrm{ctgh}}u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img875.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 0^{-}} {\\mathrm{ctgh}}u = -\\infty. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P121\" title=\"2P121\"&gt;\n                O gráfico desta função é dado por\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2307\"&gt;&lt;/a&gt;\n                &lt;table id=\"2I18\" title=\"2I18\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.18:&lt;/strong&gt;\n                Gráfico da função cotangente hiperbólica.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/fctgh.png\" alt=\"Image fctgh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P122\" title=\"2P122\"&gt;\n                A função &lt;!-- MATH\n                 $f(u) = {\\mathrm{ctgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img870.svg\" alt=\"$f(u) = {\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função ímpar e bijetora do conjunto &lt;!-- MATH\n                 $\\mathbb{R}^{*} = \\mathbb{R}-\\{0\\}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img871.svg\" alt=\"$\\mathbb{R}^{*} = \\mathbb{R}- \\{ 0 \\}$\" loading=\"lazy\"&gt;&lt;/span&gt; no conjunto &lt;!-- MATH\n                 $(-\\infty, -1) \\cup\n                (1, \\infty)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P123\" title=\"2P123\"&gt;\n                Para a função &lt;!-- MATH\n                 $w = f(u) = {\\mathrm{sech}}u = \\frac{1}{\\cosh u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img876.svg\" alt=\"$w = f(u) = {\\mathrm{sech}}u = \\frac{1}{\\cosh u}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;a name=\"2312\"&gt;&lt;/a&gt; o domínio é o conjunto dos\n                números reais, uma vez que o denominador nunca se anula, mais do que isto, o denominador é sempre maior ou igual a 1.\n                Portanto, os valores assumidos pelo quociente &lt;!-- MATH\n                 $\\frac{1}{\\cosh u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img877.svg\" alt=\"$\\frac{1}{\\cosh u}$\" loading=\"lazy\"&gt;&lt;/span&gt;, serão sempre positivos e menores ou iguais a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img250.svg\" alt=\"$1$\" loading=\"lazy\"&gt;&lt;/span&gt; e,\n                desta forma, a função é limitada inferiormente por 0 e superiormente por 1. Além disso, como cosseno hiperbólico é uma\n                função par, então a função secante também será uma função par.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P124\" title=\"2P124\"&gt;\n                À medida que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce indefinidamente, o denominador também cresce indefinidamente e, portanto,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to -\\infty} {\\mathrm{sech}}u = \\lim_{u \\to \\infty} {\\mathrm{sech}}u = 0.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P125\" title=\"2P125\"&gt;\n                &lt;img style=\"height: 3.28ex; vertical-align: -1.73ex; \" src=\"img/img878.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} {\\mathrm{sech}}u = \\lim_{u \\to \\infty} {\\mathrm{sech}}u = 0. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P126\" title=\"2P126\"&gt;\n                O gráfico desta função é\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2319\"&gt;&lt;/a&gt;\n                &lt;table id=\"2I19\" title=\"2I19\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.19:&lt;/strong&gt;\n                Gráfico da função secante hiperbólica.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/fsech.png\" alt=\"Image fsech\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P127\" title=\"2P127\"&gt;\n                A cossecante hiperbólica é dada pelo quociente &lt;!-- MATH\n                 $w = f(u) = {\\mathrm{csch}}u = \\frac{1}{{\\mathrm{senh}}u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img879.svg\" alt=\"$w = f(u) = {\\mathrm{csch}}u = \\frac{1}{{\\mathrm{senh}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;a name=\"2323\"&gt;&lt;/a&gt; e desta forma, seu domínio é o conjunto dos números reais tais que o denominador não se anula, isto\n                é, &lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;. A imagem por sua vez é também o conjunto &lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt; já que a fração &lt;!-- MATH\n                 $\\frac{1}{{\\mathrm{senh}}u}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img881.svg\" alt=\"$\\frac{1}{{\\mathrm{senh}}u}$\" loading=\"lazy\"&gt;&lt;/span&gt; jamais se anula.\n                Conforme &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; cresce (para o infinito), o denominador também cresce (para o infinito) e, assim,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to -\\infty} {\\mathrm{csch}}u = \\lim_{u \\to \\infty} {\\mathrm{csch}}u = 0.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P128\" title=\"2P128\"&gt;\n                &lt;img style=\"height: 3.28ex; vertical-align: -1.73ex; \" src=\"img/img882.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} {\\mathrm{csch}}u = \\lim_{u \\to \\infty} {\\mathrm{csch}}u = 0. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P129\" title=\"2P129\"&gt;\n                Próximo do ponto crítico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; os valores do denominador também estarão próximos de &lt;span class=\"MATH\"&gt;0&lt;/span&gt; e, portanto, fazendo o estudo\n                de sinal, temos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{u \\to 0^{+}} {\\mathrm{csch}}u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to 0^{-}} {\\mathrm{csch}}u = -\\infty.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"2P130\" title=\"2P130\"&gt;\n                &lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img883.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} {\\mathrm{csch}}u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img884.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 0^{-}} {\\mathrm{csch}}u = -\\infty. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P131\" title=\"2P131\"&gt;\n                O gráfico da função cossecante hiperbólica é\n                &lt;/p&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2335\"&gt;&lt;/a&gt;\n                &lt;table id=\"2I20\" title=\"2I20\"&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.20:&lt;/strong&gt;\n                Gráfico da função cossecante hiperbólica.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/fcsch.png\" alt=\"Image fcsch\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"2P132\" title=\"2P132\"&gt;\n                A função cossecante hiperbólica é uma função ímpar bijetora de &lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;. É decrescente em cada um dos\n                semi-eixos positivo e negativo.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"2P133\" title=\"2P133\"&gt;\n                A relação completa das funções trigonométricas hiperbólicas, com os domínios e imagens é resumida na próxima tabela.\n                \n                &lt;/p&gt;\n                &lt;br&gt;\n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"2358\"&gt;&lt;/a&gt;\n                &lt;table id=\"2T1\" title=\"2T1\"&gt;\n                &lt;caption&gt;&lt;strong&gt;Tabela 2.1:&lt;/strong&gt;\n                Domínio e imagem das funções trigonométricas hiperbólicas.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;table class=\"PAD  BORDER\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;imagem&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 ${\\mathrm{senh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;  &lt;!-- MATH\n                 $\\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;  &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                 $\\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;  &lt;!-- MATH\n                 $\\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;  &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                 $[1,\\infty)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 ${\\mathrm {tgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img864.svg\" alt=\"${\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;  &lt;!-- MATH\n                 $\\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;  &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 ${\\mathrm{ctgh}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img885.svg\" alt=\"${\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;  &lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;  &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                 $(-\\infty,-1) \\cup (1,\\infty)$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 ${\\mathrm{sech}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img886.svg\" alt=\"${\\mathrm{sech}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;  &lt;!-- MATH\n                 $\\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;  &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img887.svg\" alt=\"$(0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                 ${\\mathrm{csch}}u$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img888.svg\" alt=\"${\\mathrm{csch}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;  &lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;  &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                 $\\mathbb{R}^{*}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;",
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    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.4 Continuidade das funções trigonométricas hiperbólicas",
    "text": "2.4 Continuidade das funções trigonométricas hiperbólicas\n\n                    &lt;p class=\" unidade\" id=\"2P134\" title=\"2P134\"&gt;\n                    Agora vamos mostrar que as funções trigonométricas hiperbólicas são contínuas em cada um dos pontos de definição destas\n                    funções. Mais precisamente, mostraremos que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{x \\to a} {\\mathrm{senh}}x = {\\mathrm{senh}}a \\qquad \\text{e} \\qquad \\lim_{x \\to a} \\cosh x = \\cosh a,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P135\" title=\"2P135\"&gt;\n                    &lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img889.svg\" alt=\"$\\displaystyle \\lim_{x \\to a} {\\mathrm{senh}}x = {\\mathrm{senh}}a$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img890.svg\" alt=\"$\\displaystyle \\qquad \\lim_{x \\to a} \\cosh x = \\cosh a, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P136\" style=\"text-indent: 0 !important;\" title=\"2P136\"&gt;\n                    para qualquer &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                    \n                    &lt;/p&gt;\n                    \n                    &lt;div id=\"2Teo4\" title=\"2Teo4\" class=\" unidade\"&gt;&lt;a id=\"limfundsinh\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;O limite\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} \\frac{{\\mathrm{senh}}u}{u}\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P137\" title=\"2P137\"&gt;\n                    &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img891.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\frac{{\\mathrm{senh}}u}{u} $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P138\" style=\"text-indent: 0 !important;\" title=\"2P138\"&gt;&lt;i&gt;\n                    existe e é igual a 1.\n                    &lt;/i&gt;&lt;/p&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                    \n                    Vamos estudar os limites laterais e verificar que são ambos iguais a 1. Para obter o limite quando &lt;!-- MATH\n                     $u \\to 0^{+}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.13ex; \" src=\"img/img240.svg\" alt=\"$u \\to 0^{+}$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                    podemos considerar que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img892.svg\" alt=\"$0 &lt; u &lt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Consideremos o arco hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img62.svg\" alt=\"$AV$\" loading=\"lazy\"&gt;&lt;/span&gt;, relacionado com o ângulo hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e a\n                    reta &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\" loading=\"lazy\"&gt;&lt;/span&gt; paralela ao eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; que passa por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img310.svg\" alt=\"$V$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta reta intercepta o segmento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img100.svg\" alt=\"$OA$\" loading=\"lazy\"&gt;&lt;/span&gt; em um ponto que denominaremos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img6.svg\" alt=\"$Q$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                    (Ver figura &lt;a href=\"#figlimhip\"&gt;2.21&lt;/a&gt;).\n                    \n                    &lt;div class=\"CENTER\"&gt;&lt;a id=\"figlimhip\"&gt;&lt;/a&gt;&lt;a id=\"2375\"&gt;&lt;/a&gt;\n                    &lt;table id=\"2I21\" title=\"2I21\"&gt;\n                    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.21:&lt;/strong&gt;\n                    Visualização geométrica do limite&lt;/caption&gt;\n                    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                    &lt;div class=\"CENTER\"&gt;\n                    &lt;img src=\"img/limhip.png\" alt=\"Image limhip\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P139\" title=\"2P139\"&gt;\n                    Nestes termos, sabemos que a área do setor hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; (a área sombreada da figura (&lt;a href=\"#figanghippos\"&gt;2.5&lt;/a&gt;)) é igual\n                    a &lt;!-- MATH\n                     $\\frac{u}{2}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img61.svg\" alt=\"$\\frac{u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;, a área do triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual a &lt;!-- MATH\n                     $\\frac{{\\mathrm{senh}}u}{2}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img893.svg\" alt=\"$\\frac{{\\mathrm{senh}}u}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; e a área do triângulo retângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img894.svg\" alt=\"$OVQ$\" loading=\"lazy\"&gt;&lt;/span&gt; é igual\n                    a &lt;!-- MATH\n                     $\\frac{1}{2} {\\mathrm {tgh}}u = \\frac{1}{2} \\frac{{\\mathrm{senh}}u}{\\cosh u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img895.svg\" alt=\"$\\frac{1}{2} {\\mathrm {tgh}}u = \\frac{1}{2} \\frac{{\\mathrm{senh}}u}{\\cosh u}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também a área do triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img894.svg\" alt=\"$OVQ$\" loading=\"lazy\"&gt;&lt;/span&gt; é menor que a área do\n                    setor hiperbólico &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt; que por sua vez é menor que a área do triângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img311.svg\" alt=\"$AOV$\" loading=\"lazy\"&gt;&lt;/span&gt;, ou seja,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{{\\mathrm{senh}}u}{2 \\cosh u} &lt; \\frac{u}{2} &lt; \\frac{{\\mathrm{senh}}u}{2}.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P140\" title=\"2P140\"&gt;\n                    &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img896.svg\" alt=\"$\\displaystyle \\frac{{\\mathrm{senh}}u}{2 \\cosh u} &lt; \\frac{u}{2} &lt; \\frac{{\\mathrm{senh}}u}{2}. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P141\" title=\"2P141\"&gt;\n                    Multiplicando tudo por 2 e dividindo tudo por &lt;!-- MATH\n                     ${\\mathrm{senh}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; (que é positivo), temos\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{1}{\\cosh u} &lt; \\frac{u}{{\\mathrm{senh}}u} &lt; 1,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P142\" title=\"2P142\"&gt;\n                    &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img897.svg\" alt=\"$\\displaystyle \\frac{1}{\\cosh u} &lt; \\frac{u}{{\\mathrm{senh}}u} &lt; 1, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P143\" style=\"text-indent: 0 !important;\" title=\"2P143\"&gt;\n                    ou ainda,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\cosh u &gt; \\frac{{\\mathrm{senh}}u}{u} &gt; 1.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P144\" title=\"2P144\"&gt;\n                    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img898.svg\" alt=\"$\\displaystyle \\cosh u &gt; \\frac{{\\mathrm{senh}}u}{u} &gt; 1. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P145\" title=\"2P145\"&gt;\n                    Da primeira desigualdade temos que &lt;!-- MATH\n                     ${\\mathrm{senh}}u &lt; u \\cosh u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img899.svg\" alt=\"${\\mathrm{senh}}u &lt; u \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, usando isto, temos\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P146\" title=\"2P146\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img900.svg\" alt=\"$\\displaystyle \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.80ex; vertical-align: -0.29ex; \" src=\"img/img901.svg\" alt=\"$\\displaystyle = \\sqrt{1+{\\mathrm{senh}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.80ex; vertical-align: -0.29ex; \" src=\"img/img902.svg\" alt=\"$\\displaystyle &lt; \\sqrt{1+2{\\mathrm{senh}}u + {\\mathrm{senh}}^{2}u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img903.svg\" alt=\"$\\displaystyle = 1 + {\\mathrm{senh}}u &lt; 1 + u \\cosh u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"2P147\" title=\"2P147\"&gt;\n                    Desta forma &lt;!-- MATH\n                     $\\cosh u &lt; 1 + u \\cosh u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img904.svg\" alt=\"$\\cosh u &lt; 1 + u \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, reorganizando os termos, temos\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    (1-u) \\cosh u &lt; 1,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P148\" title=\"2P148\"&gt;\n                    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img905.svg\" alt=\"$\\displaystyle (1-u) \\cosh u &lt; 1, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P149\" style=\"text-indent: 0 !important;\" title=\"2P149\"&gt;\n                    e como &lt;!-- MATH\n                     $u \\in (0,1)$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img906.svg\" alt=\"$u \\in (0,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img907.svg\" alt=\"$(1-u) &gt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, o que nos permite obter &lt;!-- MATH\n                     $\\cosh u &lt; \\frac{1}{1-u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img908.svg\" alt=\"$\\cosh u &lt; \\frac{1}{1-u}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Segue que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    1 &lt; \\frac{{\\mathrm{senh}}u}{u} &lt; \\cosh u &lt; \\frac{1}{1-u}.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P150\" title=\"2P150\"&gt;\n                    &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img909.svg\" alt=\"$\\displaystyle 1 &lt; \\frac{{\\mathrm{senh}}u}{u} &lt; \\cosh u &lt; \\frac{1}{1-u}. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P151\" title=\"2P151\"&gt;\n                    Passando agora o limite na desigualdade, quando &lt;!-- MATH\n                     $u \\to 0^{+}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.13ex; \" src=\"img/img240.svg\" alt=\"$u \\to 0^{+}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos que o limite do termo do lado esquerdo existe e é\n                    igual a 1 e o limite do lado direito também existe e é igual a 1, pois &lt;!-- MATH\n                     $\\frac{1}{1-u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img910.svg\" alt=\"$\\frac{1}{1-u}$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função contínua em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos então pelo teorema do confronto (teorema do sanduíche) que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0^{+}} \\frac{{\\mathrm{senh}}u}{u} = 1.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P152\" title=\"2P152\"&gt;\n                    &lt;img style=\"height: 4.80ex; vertical-align: -1.82ex; \" src=\"img/img911.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} \\frac{{\\mathrm{senh}}u}{u} = 1. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P153\" title=\"2P153\"&gt;\n                    O caso em que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img52.svg\" alt=\"$u &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, é obtido observando que a função &lt;!-- MATH\n                     $\\frac{{\\mathrm{senh}}u}{u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img912.svg\" alt=\"$\\frac{{\\mathrm{senh}}u}{u}$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função par. Assim o comportamento à\n                    esquerda de 0 é o mesmo comportamento à direita de zero. Temos então\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0^{-}} \\frac{{\\mathrm{senh}}u}{u} = \\lim_{u \\to 0^{+}} \\frac{{\\mathrm{senh}}u}{u} = 1,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P154\" title=\"2P154\"&gt;\n                    &lt;img style=\"height: 4.80ex; vertical-align: -1.82ex; \" src=\"img/img913.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{-}} \\frac{{\\mathrm{senh}}u}{u} = \\lim_{u \\to 0^{+}} \\frac{{\\mathrm{senh}}u}{u} = 1, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P155\" style=\"text-indent: 0 !important;\" title=\"2P155\"&gt;\n                    e, portanto,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} \\frac{{\\mathrm{senh}}u}{u} = 1,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P156\" title=\"2P156\"&gt;\n                    &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img914.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\frac{{\\mathrm{senh}}u}{u} = 1, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    &lt;p style=\"text-indent: 0 !important; text-align: left !important;\" title=\"2P156\"&gt;&lt;span class=\"MATH\"&gt;o que encerra esta demonstração.&lt;/span&gt;&lt;span style=\"text-align: right;\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; float: right;\" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;\n                    &lt;/span&gt;&lt;/p&gt;\n                    &lt;/div&gt;\n                    \n                    \n                    &lt;div id=\"2Teo5\" title=\"2Teo5\" class=\" unidade\"&gt;&lt;a id=\"teo2.5\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;As funções seno e cosseno hiperbólico são contínuas em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} \\cosh u = 1 \\qquad \\text{e} \\qquad \\lim_{u \\to 0} {\\mathrm{senh}}u = 0.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P157\" title=\"2P157\"&gt;\n                    &lt;img style=\"height: 3.33ex; vertical-align: -1.78ex; \" src=\"img/img915.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cosh u = 1$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.33ex; vertical-align: -1.78ex; \" src=\"img/img916.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 0} {\\mathrm{senh}}u = 0.$\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                    \n                    Para o primeiro limite, supondo primeiro &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img306.svg\" alt=\"$u&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt; (mais precisamente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img892.svg\" alt=\"$0 &lt; u &lt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;), usamos a desigualdade obtida no teorema\n                    anterior\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    1 &lt; \\frac{{\\mathrm{senh}}u}{u} &lt; \\cosh u &lt; \\frac{1}{1-u},\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P158\" title=\"2P158\"&gt;\n                    &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img917.svg\" alt=\"$\\displaystyle 1 &lt; \\frac{{\\mathrm{senh}}u}{u} &lt; \\cosh u &lt; \\frac{1}{1-u}, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P159\" style=\"text-indent: 0 !important;\" title=\"2P159\"&gt;\n                    e o teorema do confronto garante que &lt;!-- MATH\n                     $\\lim\\limits_{u \\to 0^{+}} \\cosh u = 1$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img918.svg\" alt=\"$\\lim\\limits_{u \\to 0^{+}} \\cosh u = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img52.svg\" alt=\"$u &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt; lembremos que cosseno\n                    hiperbólico é uma função par e então como na demonstração do teorema anterior,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0^{-}} \\cosh u = \\lim_{u \\to 0^{+}} \\cosh u = 1,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P160\" title=\"2P160\"&gt;\n                    &lt;img style=\"height: 3.37ex; vertical-align: -1.82ex; \" src=\"img/img919.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{-}} \\cosh u = \\lim_{u \\to 0^{+}} \\cosh u = 1, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P161\" style=\"text-indent: 0 !important;\" title=\"2P161\"&gt;\n                    e isso prova o primeiro limite.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"2P162\" title=\"2P162\"&gt;\n                    Para provar o segundo limite, usaremos o item (c) do teorema &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teopop\"&gt;1.2&lt;/a&gt;. Como os limites de &lt;!-- MATH\n                     $\\frac{{\\mathrm{senh}}u}{u}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img912.svg\" alt=\"$\\frac{{\\mathrm{senh}}u}{u}$\" loading=\"lazy\"&gt;&lt;/span&gt; e de\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; existem quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; então o limite do produto existe e\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} \\frac{u{\\mathrm{senh}}u}{u} = \\lim_{u \\to 0} u \\frac{{\\mathrm{senh}}u}{u} = \\lim_{u \\to 0} u \\cdot \\lim_{u \\to 0} \\frac{{\\mathrm{senh}}u}{u} = 0 \\cdot 1 = 0.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P163\" title=\"2P163\"&gt;\n                    &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img920.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\frac{u{\\mathrm{senh}}u}{u} = \\lim_{u \\to 0} u \\fr...\n                    ...m_{u \\to 0} u \\cdot \\lim_{u \\to 0} \\frac{{\\mathrm{senh}}u}{u} = 0 \\cdot 1 = 0. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P164\" title=\"2P164\"&gt;\n                    Agora, como &lt;!-- MATH\n                     $\\frac{u {\\mathrm{senh}}u}{u} = {\\mathrm{senh}}u$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img921.svg\" alt=\"$\\frac{u {\\mathrm{senh}}u}{u} = {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img339.svg\" alt=\"$u \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; então do teorema &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teofgdif\"&gt;1.4&lt;/a&gt; segue que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} {\\mathrm{senh}}u = \\lim_{u \\to 0} \\frac{u {\\mathrm{senh}}u}{u} = 0,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P165\" title=\"2P165\"&gt;\n                    &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img922.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} {\\mathrm{senh}}u = \\lim_{u \\to 0} \\frac{u {\\mathrm{senh}}u}{u} = 0, $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    &lt;p style=\"text-indent: 0 !important; text-align: left !important;\" title=\"2P165\"&gt;&lt;span class=\"MATH\"&gt;e isso finaliza esta demonstração.&lt;/span&gt;&lt;span style=\"text-align: right;\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; float: right;\" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;\n                    &lt;/span&gt;&lt;/p&gt;\n                    &lt;/div&gt;\n                    \n                   \n                    &lt;div id=\"2Teo6\" title=\"2Teo6\" class=\" unidade\"&gt;&lt;a id=\"teo2.6\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;Para qualquer &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tem-se\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} \\cosh(u+a) = \\cosh a \\qquad \\text{e} \\qquad \\lim_{u \\to 0} {\\mathrm{senh}}(u+a) = {\\mathrm{senh}}a.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P166\" title=\"2P166\"&gt;\n                    &lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img923.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cosh(u+a) = \\cosh a$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img924.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 0} {\\mathrm{senh}}(u+a) = {\\mathrm{senh}}a.$\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                    \n                    Usando a identidade trigonométrica para a soma de arcos do cosseno hiperbólico, temos que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to 0} \\cosh(u+a) = \\lim_{u \\to 0} [\\cosh u \\cosh a + {\\mathrm{senh}}u {\\mathrm{senh}}a],\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P167\" title=\"2P167\"&gt;\n                    &lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img925.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cosh(u+a) = \\lim_{u \\to 0} [\\cosh u \\cosh a + {\\mathrm{senh}}u {\\mathrm{senh}}a], $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;\n                    e dos itens (a) e (b) do teorema &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teopop\"&gt;1.2&lt;/a&gt;, segue que\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P168\" title=\"2P168\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img926.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} \\cosh(u+a)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img927.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} [\\cosh u \\cosh a] + \\lim_{u \\to 0} [{\\mathrm{senh}}u {\\mathrm{senh}}a]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img928.svg\" alt=\"$\\displaystyle = (\\cosh a) \\lim_{u \\to 0} \\cosh u + ({\\mathrm{senh}}a) \\lim_{u \\to 0} {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img929.svg\" alt=\"$\\displaystyle = (\\cosh a) \\cdot 1 + ({\\mathrm{senh}}a) \\cdot 0 = \\cosh a.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"2P169\" title=\"2P169\"&gt;\n                    Usando agora a identidade trigonométrica para a soma de arcos do seno hiperbólico, temos\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"2P170\" title=\"2P170\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img930.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0} {\\mathrm{senh}}(u+a)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img931.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} [{\\mathrm{senh}}u \\cosh a + {\\mathrm{senh}}a \\cosh u]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img932.svg\" alt=\"$\\displaystyle = \\lim_{u \\to 0} [{\\mathrm{senh}}u \\cosh a] + \\lim_{u \\to 0} [{\\mathrm{senh}}a \\cosh u]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.45ex; vertical-align: -1.78ex; \" src=\"img/img933.svg\" alt=\"$\\displaystyle = (\\cosh a) \\lim_{u \\to 0} {\\mathrm{senh}}u + ({\\mathrm{senh}}a) \\lim_{u \\to 0} \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img934.svg\" alt=\"$\\displaystyle = (\\cosh a) \\cdot 0 + ({\\mathrm{senh}}a) \\cdot 1 = {\\mathrm{senh}}a,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    &lt;p style=\"text-indent: 0 !important; text-align: left !important;\" title=\"2P170\"&gt;&lt;span class=\"MATH\"&gt;e isso termina esta demonstração.&lt;/span&gt;&lt;span style=\"text-align: right;\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; float: right;\" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;\n                    &lt;/span&gt;&lt;/p&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P171\" title=\"2P171\"&gt;\n                    Os limites indicados no início desta seção seguem agora imediatamente do teorema de mudança de variáveis &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teomudv\"&gt;1.5&lt;/a&gt;,\n                    e dos limites que acabamos de provar.\n                    &lt;/p&gt;\n                  \n                    &lt;div id=\"2Teo7\" title=\"2Teo7\" class=\" unidade\"&gt;&lt;a id=\"cor2.7\"&gt;&lt;b&gt;Corolário  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;As funções seno e cosseno hiperbólicos são contínuas em qualquer ponto &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\lim_{u \\to a} \\cosh u = \\cosh a \\qquad \\text{e} \\qquad \\lim_{u \\to a} {\\mathrm{senh}}u = {\\mathrm{senh}}a.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"2P172\" title=\"2P172\"&gt;\n                    &lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img935.svg\" alt=\"$\\displaystyle \\lim_{u \\to a} \\cosh u = \\cosh a$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img936.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to a} {\\mathrm{senh}}u = {\\mathrm{senh}}a. $\" loading=\"lazy\"&gt;\n                    &lt;/div&gt;&lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"2P173\" title=\"2P173\"&gt;\n                    Vamos agora analisar a continuidade das outras quatro funções trigonométricas hiperbólicas, já que estas são escritas\n                    como um quociente em termos de seno e cosseno. Usando o item (d) do teorema &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teopop\"&gt;1.2&lt;/a&gt;, podemos facilmente provar as\n                    afirmações a seguir.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"2P174\" title=\"2P174\"&gt;\n                    As funções tangente, cotangente, secante e cossecante hiperbólicas são contínuas nos seus domínios de definição. Isto\n                    é,\n                    &lt;/p&gt;\n                    &lt;div class=\"CENTER\"&gt;\n                    &lt;table class=\"PAD \"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     $\\lim\\limits_{u \\to a} {\\mathrm {tgh}}u = {\\mathrm {tgh}}a$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img937.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {tgh}}u = {\\mathrm {tgh}}a$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     $\\lim\\limits_{u \\to a} {\\mathrm{ctgh}}u = {\\mathrm{ctgh}}a$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img938.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{ctgh}}u = {\\mathrm{ctgh}}a$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img939.svg\" alt=\"$a \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     $\\lim\\limits_{u \\to a} {\\mathrm{sech}}u = {\\mathrm{sech}}a$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img940.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{sech}}u = {\\mathrm{sech}}a$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n                    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                     $\\lim\\limits_{u \\to a} {\\mathrm{csch}}u = {\\mathrm{csch}}a$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.15ex; vertical-align: -1.60ex; \" src=\"img/img941.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{csch}}u = {\\mathrm{csch}}a$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img939.svg\" alt=\"$a \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/td&gt;\n                    &lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;\n                    &lt;/div&gt;",
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      "Capítulo 2: Funções trigonométricas hiperbólicas"
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    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.5 Derivadas de funções trigonométricas hiperbólicas",
    "text": "2.5 Derivadas de funções trigonométricas hiperbólicas\n\n                   &lt;a id=\"secderhip\"&gt;&lt;/a&gt;\n                     \n                        \n                        &lt;p class=\" unidade\" id=\"2P175\" title=\"2P175\"&gt;\n                        Vamos agora deduzir as derivadas das funções trigonométricas hiperbólicas. Para isto usaremos primeiro a definição de\n                        derivada, isto é,\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        f'(u) = \\lim_{h \\to 0} \\frac{f(u+h) - f(u)}{h}\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P176\" title=\"2P176\"&gt;\n                        &lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img942.svg\" alt=\"$\\displaystyle f'(u) = \\lim_{h \\to 0} \\frac{f(u+h) - f(u)}{h} $\" loading=\"lazy\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P177\" title=\"2P177\"&gt;\n                        para encontrar as derivadas de &lt;!-- MATH\n                         ${\\mathrm{senh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Depois usaremos a regra do quociente para obter as derivadas das\n                        demais funções trigonométricas hiperbólicas. Antes precisamos determinar um limite importante.\n                        \n                         &lt;/p&gt;\n                        \n                        &lt;div id=\"2Teo8\" title=\"2Teo8\" class=\" unidade\"&gt;&lt;a id=\"prop2.8\"&gt;&lt;b&gt;Proposição  &lt;span class=\"arabic\"&gt;2&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                        &lt;i&gt;O limite\n                        &lt;/i&gt;&lt;!-- MATH\n                         \\begin{displaymath}\n                        \\lim_{h \\to 0} \\frac{\\cosh h - 1}{h}\n                        \\end{displaymath}\n                         --&gt;\n                        \n                        &lt;div class=\"mathdisplay unidade\" id=\"2P178\" title=\"2P178\"&gt;\n                        &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img943.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{\\cosh h - 1}{h} $\" loading=\"lazy\"&gt;\n                        &lt;/div&gt;&lt;i&gt;\n                        existe e é igual a 0.\n                        &lt;/i&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                        \n                        Vamos modificar um pouco o quociente deste limite e usar a proposição anterior. Notemos que\n                    &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\frac{\\cosh h - 1}{h} = \\frac{\\cosh h - 1}{h} \\cdot \\frac{\\cosh h + 1}{\\cosh h +1} = \\frac{\\cosh^{2} h - 1}{h(\\cosh h + 1)}.\n                        \\end{displaymath}\n                         --&gt;\n                    \n                        &lt;div class=\"mathdisplay unidade\" id=\"2P179\" title=\"2P179\"&gt;\n                        &lt;img style=\"height: 5.50ex; vertical-align: -2.07ex; \" src=\"img/img944.svg\" alt=\"$\\displaystyle \\frac{\\cosh h - 1}{h} = \\frac{\\cosh h - 1}{h} \\cdot \\frac{\\cosh h + 1}{\\cosh h +1} = \\frac{\\cosh^{2} h - 1}{h(\\cosh h + 1)}. $\" loading=\"lazy\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"2P180\" title=\"2P180\"&gt;\n                        Usando agora a identidade fundamental (&lt;a href=\"#idfundhip\"&gt;2.3&lt;/a&gt;) no membro da direita, temos que\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\frac{\\cosh h - 1}{h} = \\frac{{\\mathrm{senh}}^{2} h}{h(\\cosh h + 1)} = \\frac{{\\mathrm{senh}}h}{h} \\cdot \\frac{{\\mathrm{senh}}h}{\\cosh h + 1}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P181\" title=\"2P181\"&gt;\n                        &lt;img style=\"height: 5.50ex; vertical-align: -2.07ex; \" src=\"img/img945.svg\" alt=\"$\\displaystyle \\frac{\\cosh h - 1}{h} = \\frac{{\\mathrm{senh}}^{2} h}{h(\\cosh h + 1)} = \\frac{{\\mathrm{senh}}h}{h} \\cdot \\frac{{\\mathrm{senh}}h}{\\cosh h + 1}. $\" loading=\"lazy\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"2P182\" title=\"2P182\"&gt;\n                        Olhando para o membro da direira, temos que, o limite da primeira fração quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img946.svg\" alt=\"$h \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; existe e é igual a 1\n                        (proposição &lt;a href=\"#limfundsinh\"&gt;2.4&lt;/a&gt;) e o limite da segunda fração quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img946.svg\" alt=\"$h \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; também existe por ser uma função contínua\n                        em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img293.svg\" alt=\"$h$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta forma o limite do produto existe quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img946.svg\" alt=\"$h \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e,\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P183\" title=\"2P183\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img943.svg\" alt=\"$\\displaystyle \\lim_{h \\to 0} \\frac{\\cosh h - 1}{h} $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img947.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}h}{h} \\cdot \\frac{{\\mathrm{senh}}h}{\\cosh h + 1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img948.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}h}{h} \\cdot \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}h}{\\cosh h + 1} = 1 \\cdot \\left( \\frac{0}{1+1} \\right) = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        &lt;p style=\"text-indent: 0 !important; text-align: left !important;\" title=\"2P183\"&gt;&lt;span class=\"MATH\"&gt;e a prova está terminada.&lt;/span&gt;&lt;span style=\"text-align: right;\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; float: right;\" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;\n                        &lt;/span&gt;&lt;/p&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"2P184\" title=\"2P184\"&gt;\n                        Agora temos condições de deduzir as fórmulas de derivada para as funções trigonométricas seno e cosseno\n                        hiperbólicos. Para a função seno hiperbólico temos que a derivada é dada por, &lt;a name=\"2552\"&gt;&lt;/a&gt;\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\frac{d}{du} {\\mathrm{senh}}u = \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}(u+h) - {\\mathrm{senh}}u}{h}\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P185\" title=\"2P185\"&gt;\n                        &lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img949.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{senh}}u = \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}(u+h) - {\\mathrm{senh}}u}{h} $\" loading=\"lazy\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P186\" style=\"text-indent: 0 !important;\" title=\"2P186\"&gt;\n                        em todos os valores &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que o limite existe.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"2P187\" title=\"2P187\"&gt;\n                        Assim,\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P188\" title=\"2P188\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img950.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img951.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}(u+h) - {\\mathrm{senh}}u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img952.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}u \\cosh h + {\\mathrm{senh}}h \\cosh u - {\\mathrm{senh}}u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img953.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\left[ {\\mathrm{senh}}u \\frac{(\\cosh h - 1)}{h} + \\frac{{\\mathrm{senh}}h}{h} \\cosh u \\right]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"2P189\" style=\"text-indent: 0 !important;\" title=\"2P189\"&gt;\n                        para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que o limite acima existe.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"2P190\" title=\"2P190\"&gt;\n                        Mas os limites de &lt;!-- MATH\n                         ${\\mathrm{senh}}u \\frac{(\\cosh h - 1)}{h}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.11ex; vertical-align: -0.83ex; \" src=\"img/img954.svg\" alt=\"${\\mathrm{senh}}u \\frac{(\\cosh h - 1)}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                         $\\frac{{\\mathrm{senh}}h}{h} \\cosh u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img955.svg\" alt=\"$\\frac{{\\mathrm{senh}}h}{h} \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; existem para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e assim,\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P191\" title=\"2P191\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img950.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img953.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\left[ {\\mathrm{senh}}u \\frac{(\\cosh h - 1)}{h} + \\frac{{\\mathrm{senh}}h}{h} \\cosh u \\right]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img956.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}u \\left( \\lim_{h \\to 0} \\frac{\\cosh h - 1}{h} \\r...\n                        ...) + \\cosh u \\left( \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}h}{h} \\right)\n                        = \\cosh u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"2P192\" style=\"text-indent: 0 !important;\" title=\"2P192\"&gt;\n                        para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"2P193\" title=\"2P193\"&gt;\n                        Para a função cosseno hiperbólico, temos que &lt;a name=\"2592\"&gt;&lt;/a&gt;\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\frac{d}{du} \\cosh u = \\lim_{h \\to 0} \\frac{\\cosh(u+h) - \\cosh u}{h}\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P194\" title=\"2P194\"&gt;\n                        &lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img957.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cosh u = \\lim_{h \\to 0} \\frac{\\cosh(u+h) - \\cosh u}{h} $\" loading=\"lazy\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P195\" style=\"text-indent: 0 !important;\" title=\"2P195\"&gt;\n                        para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que o limite exista. Para tais &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P196\" title=\"2P196\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img958.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.78ex; \" src=\"img/img959.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{\\cosh(u+h) - \\cosh u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img960.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\frac{\\cosh u \\cosh h + {\\mathrm{senh}}h {\\mathrm{senh}}u - \\cosh u}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img961.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\left[ \\cosh u \\frac{(\\cosh h - 1)}{h} + \\frac{{\\mathrm{senh}}h}{h} {\\mathrm{senh}}u \\right].$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"2P197\" title=\"2P197\"&gt;\n                        Como os limites de cada uma das frações &lt;!-- MATH\n                         $\\cosh u \\frac{(\\cosh h - 1)}{h}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.11ex; vertical-align: -0.83ex; \" src=\"img/img962.svg\" alt=\"$\\cosh u \\frac{(\\cosh h - 1)}{h}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                         $\\frac{{\\mathrm{senh}}h}{h} {\\mathrm{senh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img963.svg\" alt=\"$\\frac{{\\mathrm{senh}}h}{h} {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; existem para\n                        todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; então\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P198\" title=\"2P198\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img958.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img964.svg\" alt=\"$\\displaystyle = \\lim_{h \\to 0} \\left[ \\cosh u \\frac{(\\cosh h - 1)}{h} + \\frac{{\\mathrm{senh}}h}{h} {\\mathrm{senh}}u \\right]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img965.svg\" alt=\"$\\displaystyle = \\cosh u \\left( \\lim_{h \\to 0} \\frac{\\cosh h - 1}{h} \\right)+ {\\...\n                        ...}u \\left( \\lim_{h \\to 0} \\frac{{\\mathrm{senh}}h}{h} \\right) =\n                        {\\mathrm{senh}}u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"2P199\" style=\"text-indent: 0 !important;\" title=\"2P199\"&gt;\n                        para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"2P200\" title=\"2P200\"&gt;\n                        Para as demais funções trigonométricas hiperbólicas usaremos as identidades em termos de seno e cosseno hiperbólico e a\n                        regra de derivação do quociente. Já que as funções seno e cosseno hiperólico são diferenciáveis em todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;\n                        então os quocientes de definição das demais funções trigonométricas hiperbólicas são diferenciáveis em todos os pontos\n                        onde o denominador não se anula.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"2P201\" title=\"2P201\"&gt;\n                        A função &lt;!-- MATH\n                         $f(u) = {\\mathrm {tgh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img860.svg\" alt=\"$f(u) = {\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; é diferenciável em todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;a name=\"2632\"&gt;&lt;/a&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P202\" title=\"2P202\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img966.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {tgh}}u = \\frac{d}{du} \\left( \\tfrac{{\\mathrm{senh}}u}{\\cosh u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.82ex; \" src=\"img/img967.svg\" alt=\"$\\displaystyle = \\frac{({\\mathrm{senh}}u)' \\cosh u - {\\mathrm{senh}}u (\\cosh u)'}{\\cosh^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.79ex; vertical-align: -1.82ex; \" src=\"img/img968.svg\" alt=\"$\\displaystyle = \\frac{\\cosh u \\cosh u - {\\mathrm{senh}}u {\\mathrm{senh}}u}{\\cosh^{2} u} = \\frac{1}{\\cosh^{2} u} = {\\mathrm{sech}}^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"2P203\" title=\"2P203\"&gt;\n                        Para a função cotangente, temos em todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}^{*}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.75ex; vertical-align: -0.18ex; \" src=\"img/img969.svg\" alt=\"$u \\in \\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;a name=\"2649\"&gt;&lt;/a&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P204\" title=\"2P204\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img970.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{ctgh}}u = \\frac{d}{du} \\left( \\tfrac{\\cosh u}{{\\mathrm{senh}}u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.82ex; \" src=\"img/img971.svg\" alt=\"$\\displaystyle = \\frac{(\\cosh u)' {\\mathrm{senh}}u - \\cosh u ({\\mathrm{senh}}u)'}{{\\mathrm{senh}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.79ex; vertical-align: -1.82ex; \" src=\"img/img972.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm{senh}}u {\\mathrm{senh}}u - \\cosh u \\cosh u}{{\\mathrm{senh}}^{2} u} = \\frac{-1}{{\\mathrm{senh}}^{2} u} = -{\\mathrm{csch}}^{2} u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"2P205\" title=\"2P205\"&gt;\n                        E, finalmente, para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;a name=\"2665\"&gt;&lt;/a&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P206\" title=\"2P206\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img973.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{sech}}u = \\frac{d}{du} \\left( \\tfrac{1}{\\cosh u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.82ex; \" src=\"img/img974.svg\" alt=\"$\\displaystyle = \\frac{(1)' \\cosh u - {\\mathrm{senh}}u}{\\cosh^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.79ex; vertical-align: -1.82ex; \" src=\"img/img975.svg\" alt=\"$\\displaystyle = \\frac{-{\\mathrm{senh}}u}{\\cosh^{2} u} = - \\frac{1}{\\cosh u} \\frac{{\\mathrm{senh}}u}{\\cosh u} = -{\\mathrm{sech}}u {\\mathrm {tgh}}u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"2P207\" style=\"text-indent: 0 !important;\" title=\"2P207\"&gt;\n                        e também, para todo &lt;!-- MATH\n                         $u \\in \\mathbb{R}^{*}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.75ex; vertical-align: -0.18ex; \" src=\"img/img969.svg\" alt=\"$u \\in \\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;a name=\"2683\"&gt;&lt;/a&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"2P208\" title=\"2P208\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img976.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{csch}}u = \\frac{d}{du} \\left( \\tfrac{1}{{\\mathrm{senh}}u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.95ex; vertical-align: -1.82ex; \" src=\"img/img977.svg\" alt=\"$\\displaystyle = \\frac{(1)'{\\mathrm{senh}}u - \\cosh u}{{\\mathrm{senh}}^{2} u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.79ex; vertical-align: -1.82ex; \" src=\"img/img978.svg\" alt=\"$\\displaystyle = \\frac{-\\cosh u}{{\\mathrm{senh}}^{2} u} = - \\frac{1}{{\\mathrm{senh}}u} \\frac{\\cosh u}{{\\mathrm{senh}}u} = -{\\mathrm{csch}}u {\\mathrm{ctgh}}u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"2P209\" title=\"2P209\"&gt;\n                        A tabela abaixo, reúne as fórmulas de derivação para as funções trigonométricas hiperbólicas. O conjunto domínio\n                        descrito na tabela é o domínio da derivada. Note a semelhança com o caso circular.\n                        \n                        &lt;/p&gt;\n                        &lt;br&gt;\n                        &lt;div class=\"CENTER\"&gt;&lt;a id=\"2710\"&gt;&lt;/a&gt;\n                        &lt;table id=\"2T2\" title=\"2T2\"&gt;\n                        &lt;caption&gt;&lt;strong&gt;Tabela 2.2:&lt;/strong&gt;\n                        Derivadas das funções trigonométricas hiperbólicas.&lt;/caption&gt;\n                        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                        &lt;div class=\"CENTER\"&gt;\n                        &lt;table class=\"PAD  BORDER\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;derivada&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                         ${\\mathrm{senh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $\\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img859.svg\" alt=\"$\\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $\\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         ${\\mathrm{senh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img847.svg\" alt=\"${\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                         ${\\mathrm {tgh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img864.svg\" alt=\"${\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $\\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         ${\\mathrm{sech}}^{2} u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img979.svg\" alt=\"${\\mathrm{sech}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                         ${\\mathrm{ctgh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img885.svg\" alt=\"${\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $\\mathbb{R}^{*}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $-{\\mathrm{csch}}^{2} u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.28ex; vertical-align: -0.27ex; \" src=\"img/img980.svg\" alt=\"$-{\\mathrm{csch}}^{2} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                         ${\\mathrm{sech}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img886.svg\" alt=\"${\\mathrm{sech}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $\\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $-{\\mathrm{sech}}u {\\mathrm {tgh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img981.svg\" alt=\"$-{\\mathrm{sech}}u {\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n                         ${\\mathrm{csch}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img888.svg\" alt=\"${\\mathrm{csch}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n                         $\\mathbb{R}^{*}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n                        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n                         $-{\\mathrm{csch}}u {\\mathrm{ctgh}}u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img982.svg\" alt=\"$-{\\mathrm{csch}}u {\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;\n                        &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;\n                        &lt;/div&gt;",
    "crumbs": [
      "Capítulo 2: Funções trigonométricas hiperbólicas"
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    "href": "funcoes-trigonometricas-hiperbolicas.html#SECTION00660000000000000000",
    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.6 Funções trigonométricas hiperbólicas inversas",
    "text": "2.6 Funções trigonométricas hiperbólicas inversas\n\n&lt;p class=\" unidade\" id=\"2P210\" title=\"2P210\"&gt;\nNesta seção, vamos definir as funções trigonométricas inversas, estabelecendo os domínios, as imagens e indicando\nalguns limites importantes. Também apresentaremos os gráficos destas funções. Este não é um trabalho muito fácil pois,\ncomo acabamos de ver, as funções trigonométricas hiperbólicas não são todas elas bijetoras. Já passamos por este\nproblema na seção &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#secfcinv\"&gt;1.5&lt;/a&gt; com as funções trigonométricas circulares. Vamos impor, quando necessário, condições\nde restrição de domínio e de imagem para tornar as funções bijetivas.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"2P211\" title=\"2P211\"&gt;\nComecemos com a função seno hiperbólico, que como vimos anteriormente, é uma função bijetora de &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta\nforma, podemos obter a função inversa do seno hiperbólico, para qualquer valor real. Dado &lt;!-- MATH\n $u \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, o seno\nhiperbólico inverso de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, é o número &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt;, representado por &lt;!-- MATH\n $w = {\\mathrm{senh}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img983.svg\" alt=\"$w = {\\mathrm{senh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, que satisfaz &lt;!-- MATH\n $u = {\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img984.svg\" alt=\"$u = {\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. É usual\nrepresentar também a função seno hiperbólico inverso por &lt;!-- MATH\n $w = \\operatorname{arcsinh} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img985.svg\" alt=\"$w = \\operatorname{arcsinh} u$\" loading=\"lazy\"&gt;&lt;/span&gt; e lemos “arco seno\nhiperbólico”. Vamos usar neste texto a primeira notação e lembre-se de não confundir &lt;!-- MATH\n ${\\mathrm{senh}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img986.svg\" alt=\"${\\mathrm{senh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n $({\\mathrm{senh}}\nu)^{-1}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img987.svg\" alt=\"$({\\mathrm{senh}}\nu)^{-1}$\" loading=\"lazy\"&gt;&lt;/span&gt;. A segunda expressão é o inverso multiplicativo do seno hiperbólico, ou seja a cossecante hiperbólica.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"2P212\" title=\"2P212\"&gt;\nFazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; variar em &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos a função seno hiperbólico inverso, &lt;a name=\"2718\"&gt;&lt;/a&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P213\" title=\"2P213\"&gt;\n&lt;!-- MATH\n \\begin{eqnarray*}\nf : \\mathbb{R}& \\to & \\mathbb{R}\\\\\nu & \\mapsto & w = f(u) = {\\mathrm{senh}}^{-1} u,\n\\end{eqnarray*}\n --&gt;\n&lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n&lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img210.svg\" alt=\"$\\displaystyle f : \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img212.svg\" alt=\"$\\displaystyle \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img988.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm{senh}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;br clear=\"ALL\"&gt;\n&lt;p class=\" unidade\" id=\"2P214\" style=\"text-indent: 0 !important;\" title=\"2P214\"&gt;\nque satisfaz a relação &lt;!-- MATH\n $u = {\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img984.svg\" alt=\"$u = {\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se fizermos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; tender para o infinito, a relação &lt;!-- MATH\n $u = {\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img984.svg\" alt=\"$u = {\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; nos diz que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt;\ntambém deve ir para o infinito e analogamente para &lt;!-- MATH\n $u \\to -\\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img865.svg\" alt=\"$u \\to -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos assim,\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{u \\to \\infty} {\\mathrm{senh}}^{-1} u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to -\\infty} {\\mathrm{senh}}^{-1} u = -\\infty.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P215\" title=\"2P215\"&gt;\n&lt;img style=\"height: 3.63ex; vertical-align: -1.60ex; \" src=\"img/img989.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\infty} {\\mathrm{senh}}^{-1} u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.76ex; vertical-align: -1.73ex; \" src=\"img/img990.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to -\\infty} {\\mathrm{senh}}^{-1} u = -\\infty. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P216\" title=\"2P216\"&gt;\nValem as seguintes relações inversas,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P217\" title=\"2P217\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img991.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}({\\mathrm{senh}}^{-1} u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img992.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img993.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}^{-1}({\\mathrm{senh}}u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img994.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P218\" title=\"2P218\"&gt;\nO gráfico da função seno hiperbólico inverso, é da forma,\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2735\"&gt;&lt;/a&gt;\n&lt;table id=\"2I22\" title=\"2I22\"&gt;\n&lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.22:&lt;/strong&gt;\nGráfico de seno hiperbólico inverso.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;img src=\"img/farcsinh.png\" alt=\"Image farcsinh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P219\" title=\"2P219\"&gt;\nA função cosseno hiperbólico não é uma função bijetora. Lembremos que seu domínio é &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, mas sua imagem é o\nsubconjunto &lt;!-- MATH\n $[1,\\infty) \\subset \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img995.svg\" alt=\"$[1,\\infty) \\subset \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Restringindo o contradomínio a &lt;!-- MATH\n $[1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; tornamos esta função sobrejetora.\nTambém a função cosseno hiperbólico, definida em todo o domínio &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, não é injetora. Vamos então restringir o domínio\ndesta função ao conjunto dos reais não negativos. Temos assim que a função cosseno hiperbólico é bijetora de\n&lt;!-- MATH\n $[0,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img996.svg\" alt=\"$[0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;!-- MATH\n $[1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Por restrição, podemos então definir a função cosseno hiperbólico inverso, denotada por\n&lt;a name=\"2737\"&gt;&lt;/a&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P220\" title=\"2P220\"&gt;\n&lt;!-- MATH\n \\begin{eqnarray*}\nf : [1,\\infty) & \\to & [0,\\infty) \\\\\nu & \\mapsto & w = f(u) = \\cosh^{-1} u,\n\\end{eqnarray*}\n --&gt;\n&lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n&lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img997.svg\" alt=\"$\\displaystyle f : [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img998.svg\" alt=\"$\\displaystyle [0,\\infty)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img999.svg\" alt=\"$\\displaystyle w = f(u) = \\cosh^{-1} u,$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;br clear=\"ALL\"&gt;\n&lt;p class=\" unidade\" id=\"2P221\" style=\"text-indent: 0 !important;\" title=\"2P221\"&gt;\ne que satisfaz a relação &lt;!-- MATH\n $u = \\cosh w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1000.svg\" alt=\"$u = \\cosh w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Levando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; ao infinito, a relação &lt;!-- MATH\n $u = \\cosh w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1000.svg\" alt=\"$u = \\cosh w$\" loading=\"lazy\"&gt;&lt;/span&gt; nos mostra que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; também vai\npara o infinito. No outro extremo do intervalo de definição, isto é, quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; tende para 1 (somente pela direita), a\nmesma relação mostra que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; vai para 0. Então,\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{u \\to \\infty} \\cosh^{-1} u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to 1^{+}} \\cosh^{-1} u = 0.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P222\" title=\"2P222\"&gt;\n&lt;img style=\"height: 3.63ex; vertical-align: -1.60ex; \" src=\"img/img1001.svg\" alt=\"$\\displaystyle \\lim_{u \\to \\infty} \\cosh^{-1} u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.84ex; vertical-align: -1.81ex; \" src=\"img/img1002.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 1^{+}} \\cosh^{-1} u = 0. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P223\" title=\"2P223\"&gt;\nO gráfico desta função é a curva da figura abaixo.\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2748\"&gt;&lt;/a&gt;\n&lt;table id=\"2I23\" title=\"2I23\"&gt;\n&lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.23:&lt;/strong&gt;\nGráfico da função cosseno hiperbólico inverso.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;img src=\"img/farccosh.png\" alt=\"Image farccosh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P224\" title=\"2P224\"&gt;\nOcorrem as seguintes relações inversas,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P225\" title=\"2P225\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1003.svg\" alt=\"$\\displaystyle \\cosh(\\cosh^{-1} u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1004.svg\" alt=\"$\\displaystyle \\quad u \\in [1,\\infty),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1005.svg\" alt=\"$\\displaystyle \\cosh^{-1}(\\cosh u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1006.svg\" alt=\"$\\displaystyle \\quad u \\in [0,\\infty).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P226\" title=\"2P226\"&gt;\nA função tangente hiperbólica é uma função injetora do conjunto &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; no conjunto &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, mas não é sobrejetora já que o\nconjunto imagem é o intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Restringindo o contradomínio temos a bijetividade da função tangente\nhiperbólica de &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Definimos então a função tangente hiperbólica inversa, &lt;a name=\"2756\"&gt;&lt;/a&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P227\" title=\"2P227\"&gt;\n&lt;!-- MATH\n \\begin{eqnarray*}\nf : (-1,1) & \\to & \\mathbb{R}\\\\\nu & \\mapsto & w = f(u) = {\\mathrm {tgh}}^{-1} u,\n\\end{eqnarray*}\n --&gt;\n&lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n&lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1007.svg\" alt=\"$\\displaystyle f : (-1,1)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img212.svg\" alt=\"$\\displaystyle \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1008.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm {tgh}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;br clear=\"ALL\"&gt;\n&lt;p class=\" unidade\" id=\"2P228\" style=\"text-indent: 0 !important;\" title=\"2P228\"&gt;\ncom &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; satisfazendo &lt;!-- MATH\n $u = {\\mathrm {tgh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1009.svg\" alt=\"$u = {\\mathrm {tgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Vamos observar o seu comportamento nos extremos do intervalo. Quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; tende\na 1 (pela esquerda) então a relação &lt;!-- MATH\n $u = {\\mathrm {tgh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1009.svg\" alt=\"$u = {\\mathrm {tgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; mostra que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir para o infinito. Analogamente se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1010.svg\" alt=\"$u \\to -1$\" loading=\"lazy\"&gt;&lt;/span&gt;\nentão &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; vai para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img1011.svg\" alt=\"$-\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;. Resumindo,\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{u \\to 1^{-}} {\\mathrm {tgh}}^{-1} u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to -1^{+}} {\\mathrm {tgh}}^{-1} u = -\\infty.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P229\" title=\"2P229\"&gt;\n&lt;img style=\"height: 3.84ex; vertical-align: -1.81ex; \" src=\"img/img1012.svg\" alt=\"$\\displaystyle \\lim_{u \\to 1^{-}} {\\mathrm {tgh}}^{-1} u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.96ex; vertical-align: -1.93ex; \" src=\"img/img1013.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to -1^{+}} {\\mathrm {tgh}}^{-1} u = -\\infty. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P230\" title=\"2P230\"&gt;\nO gráfico da função tangente hiperbólica inversa,\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2767\"&gt;&lt;/a&gt;\n&lt;table id=\"2I24\" title=\"2I24\"&gt;\n&lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.24:&lt;/strong&gt;\nGráfico da função tangente hiperbólica inversa. &lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;img src=\"img/farctgh.png\" alt=\"Image farctgh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P231\" title=\"2P231\"&gt;\nAs relações inversas são\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P232\" title=\"2P232\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1014.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}({\\mathrm {tgh}}^{-1} u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1015.svg\" alt=\"$\\displaystyle \\quad u \\in (-1,1),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1016.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}^{-1}({\\mathrm {tgh}}u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img994.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P233\" title=\"2P233\"&gt;\nA função cotangente hiperbólica também é uma função bijetora do conjunto &lt;!-- MATH\n $\\mathbb{R}^{*}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt; no conjunto &lt;!-- MATH\n $(-\\infty,-1) \\cup\n(1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta forma, definimos a função cotangente hiperbólica inversa por, &lt;a name=\"2776\"&gt;&lt;/a&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P234\" title=\"2P234\"&gt;\n&lt;!-- MATH\n \\begin{eqnarray*}\nf : (-\\infty,-1) \\cup (1,\\infty) & \\to & \\mathbb{R}-\\{0\\} \\\\\nu & \\mapsto & w = f(u) = {\\mathrm{ctgh}}^{-1} u,\n\\end{eqnarray*}\n --&gt;\n&lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n&lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1017.svg\" alt=\"$\\displaystyle f : (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1018.svg\" alt=\"$\\displaystyle \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1019.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm{ctgh}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;br clear=\"ALL\"&gt;\n&lt;p class=\" unidade\" id=\"2P235\" style=\"text-indent: 0 !important;\" title=\"2P235\"&gt;\ndesde que &lt;!-- MATH\n $u = {\\mathrm{ctgh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1020.svg\" alt=\"$u = {\\mathrm{ctgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Analisando os extremos do intervalo de definição, temos que quando &lt;!-- MATH\n $u \\to -\\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img865.svg\" alt=\"$u \\to -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; a relação\n&lt;!-- MATH\n $u = {\\mathrm{ctgh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1020.svg\" alt=\"$u = {\\mathrm{ctgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; nos diz que isto ocorre quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; vai para 0 (com valores negativos). Analogamente, quando &lt;!-- MATH\n $u \\to\n\\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img863.svg\" alt=\"$u \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; então deve ocorrer &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1021.svg\" alt=\"$w \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; (com valores positivos). Fazendo &lt;!-- MATH\n $u \\to -1^{-}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.03ex; vertical-align: -0.27ex; \" src=\"img/img1022.svg\" alt=\"$u \\to -1^{-}$\" loading=\"lazy\"&gt;&lt;/span&gt; então, a mesma relação anterior,\nnos diz que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img1011.svg\" alt=\"$-\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; e analogamente &lt;!-- MATH\n $w \\to \\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1023.svg\" alt=\"$w \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; quando &lt;!-- MATH\n $u \\to 1^{+}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.86ex; vertical-align: -0.11ex; \" src=\"img/img1024.svg\" alt=\"$u \\to 1^{+}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Resumindo,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P236\" title=\"2P236\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.76ex; vertical-align: -1.73ex; \" src=\"img/img1025.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} {\\mathrm{ctgh}}^{-1} u = \\lim_{u \\to \\infty} {\\mathrm{ctgh}}^{-1} u = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.96ex; vertical-align: -1.93ex; \" src=\"img/img1026.svg\" alt=\"$\\displaystyle \\lim_{u \\to -1^{-}} {\\mathrm{ctgh}}^{-1} u = -\\infty,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.84ex; vertical-align: -1.81ex; \" src=\"img/img1027.svg\" alt=\"$\\displaystyle \\lim_{u \\to 1^{+}} {\\mathrm{ctgh}}^{-1} u = \\infty.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P237\" title=\"2P237\"&gt;\nO gráfico desta função é dado por\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2794\"&gt;&lt;/a&gt;\n&lt;table id=\"2I25\" title=\"2I25\"&gt;\n&lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.25:&lt;/strong&gt;\nGráfico da função cotangente hiperbólica inversa.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;img src=\"img/farcctgh.png\" alt=\"Image farcctgh\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P238\" title=\"2P238\"&gt;\nValem as seguintes relações de inversão,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P239\" title=\"2P239\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1028.svg\" alt=\"$\\displaystyle {\\mathrm{ctgh}}({\\mathrm{ctgh}}^{-1} u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1029.svg\" alt=\"$\\displaystyle \\quad u \\in (-\\infty,-1) \\cup (1,\\infty),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1030.svg\" alt=\"$\\displaystyle {\\mathrm{ctgh}}^{-1}({\\mathrm{ctgh}}u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1031.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}-\\{0\\}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P240\" title=\"2P240\"&gt;\nPara a secante hiperbólica, temos alguns problemas como no caso do cosseno hiperbólico inverso. O domínio da\nfunção secante hiperbólica é o conjunto &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e a imagem é o conjunto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img887.svg\" alt=\"$(0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mas esta função não é injetora de &lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;\nem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img887.svg\" alt=\"$(0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então vamos restringir o conjunto domínio para os reais não negativos. Assim, a função secante hiperbólica\né bijetiva de &lt;!-- MATH\n $[0,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img996.svg\" alt=\"$[0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img887.svg\" alt=\"$(0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; e podemos definir a função secante hiperbólica inversa &lt;a name=\"2802\"&gt;&lt;/a&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P241\" title=\"2P241\"&gt;\n&lt;!-- MATH\n \\begin{eqnarray*}\nf: (0,1] & \\to & [0,\\infty) \\\\\nu & \\mapsto & w = f(u) = {\\mathrm{sech}}^{-1} u,\n\\end{eqnarray*}\n --&gt;\n&lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n&lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1032.svg\" alt=\"$\\displaystyle f: (0,1]$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img998.svg\" alt=\"$\\displaystyle [0,\\infty)$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1033.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm{sech}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;br clear=\"ALL\"&gt;\n&lt;p class=\" unidade\" id=\"2P242\" style=\"text-indent: 0 !important;\" title=\"2P242\"&gt;\nsatisfazendo &lt;!-- MATH\n $u = {\\mathrm{sech}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1034.svg\" alt=\"$u = {\\mathrm{sech}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; (pela direita), a relação &lt;!-- MATH\n $u = {\\mathrm{sech}}w = \\frac{1}{\\cosh w}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1035.svg\" alt=\"$u = {\\mathrm{sech}}w = \\frac{1}{\\cosh w}$\" loading=\"lazy\"&gt;&lt;/span&gt; diz que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1036.svg\" alt=\"$\\cosh\nw$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar indo para o infinito por valores positivos e consequentemente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar indo para o infinito.\nQuando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; vai para 1 (pela esquerda) então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1036.svg\" alt=\"$\\cosh\nw$\" loading=\"lazy\"&gt;&lt;/span&gt; está indo para 1 e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve estar se aproximando de 0. Temos\nentão\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{u \\to 0^{+}} {\\mathrm{sech}}^{-1} u = \\infty \\qquad \\text{e} \\qquad \\lim_{u \\to 1^{-}} {\\mathrm{sech}}^{-1} u = 0.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P243\" title=\"2P243\"&gt;\n&lt;img style=\"height: 3.85ex; vertical-align: -1.82ex; \" src=\"img/img1037.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} {\\mathrm{sech}}^{-1} u = \\infty$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 3.84ex; vertical-align: -1.81ex; \" src=\"img/img1038.svg\" alt=\"$\\displaystyle \\qquad \\lim_{u \\to 1^{-}} {\\mathrm{sech}}^{-1} u = 0. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P244\" title=\"2P244\"&gt;\nAs relações inversas ficam,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P245\" title=\"2P245\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1039.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}({\\mathrm{sech}}^{-1} u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1040.svg\" alt=\"$\\displaystyle \\quad u \\in (0,1],$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1041.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}^{-1}({\\mathrm{sech}}u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1006.svg\" alt=\"$\\displaystyle \\quad u \\in [0,\\infty).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P246\" title=\"2P246\"&gt;\nGraficamente, temos\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2821\"&gt;&lt;/a&gt;\n&lt;table id=\"2I26\" title=\"2I26\"&gt;\n&lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.26:&lt;/strong&gt;\nGráfico de secante hiperbólica inversa.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;img src=\"img/farcsech.png\" alt=\"Image farcsech\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P247\" title=\"2P247\"&gt;\nFinalmente, lembremos que a função cossecante hiperbólica é bijetora do conjunto &lt;!-- MATH\n $\\mathbb{R}^{*}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt; no conjunto &lt;!-- MATH\n $\\mathbb{R}^{*}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\nDefinimos então a função cossecante hiperbólica inversa &lt;a name=\"2825\"&gt;&lt;/a&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P248\" title=\"2P248\"&gt;\n&lt;!-- MATH\n \\begin{eqnarray*}\nf: \\mathbb{R}-\\{0\\} & \\to & \\mathbb{R}-\\{0\\} \\\\\nu & \\mapsto & w = f(u) = {\\mathrm{csch}}^{-1} u,\n\\end{eqnarray*}\n --&gt;\n&lt;table cellpadding=\"0\" align=\"CENTER\" width=\"100%\"&gt;\n&lt;tbody&gt;&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1042.svg\" alt=\"$\\displaystyle f: \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img211.svg\" alt=\"$\\displaystyle \\to$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1018.svg\" alt=\"$\\displaystyle \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr valign=\"MIDDLE\"&gt;&lt;td nowrap=\"\" width=\"50%\" align=\"RIGHT\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img213.svg\" alt=\"$\\displaystyle u$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td width=\"10\" align=\"CENTER\" nowrap=\"\"&gt;&lt;img style=\"height: 0.97ex; vertical-align: -0.10ex; \" src=\"img/img214.svg\" alt=\"$\\displaystyle \\mapsto$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td align=\"LEFT\" nowrap=\"\" width=\"50%\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1043.svg\" alt=\"$\\displaystyle w = f(u) = {\\mathrm{csch}}^{-1} u,$\" loading=\"lazy\"&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" width=\"10\" align=\"RIGHT\"&gt;\n&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;br clear=\"ALL\"&gt;\n&lt;p class=\" unidade\" id=\"2P249\" style=\"text-indent: 0 !important;\" title=\"2P249\"&gt;\nque também satisfaz &lt;!-- MATH\n $u = {\\mathrm{csch}}w = \\frac{1}{{\\mathrm{senh}}w}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1044.svg\" alt=\"$u = {\\mathrm{csch}}w = \\frac{1}{{\\mathrm{senh}}w}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta relação explica também os limites. Quando &lt;!-- MATH\n $u \\to -\\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img865.svg\" alt=\"$u \\to -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;\nentão &lt;!-- MATH\n ${\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1045.svg\" alt=\"${\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir para 0 por valores negativos e então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir para 0 também por valores negativos.\nAnalogamente, quando &lt;!-- MATH\n $u \\to \\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img863.svg\" alt=\"$u \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;!-- MATH\n ${\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1045.svg\" alt=\"${\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir para 0 por valores positivos e então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir também para 0\npor valores positivos. Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; por valores positivos então &lt;!-- MATH\n ${\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1045.svg\" alt=\"${\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; deve ir para o infinito e &lt;!-- MATH\n $w \\to \\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1023.svg\" alt=\"$w \\to \\infty$\" loading=\"lazy\"&gt;&lt;/span&gt;\ntambém. Da mesma forma, se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img270.svg\" alt=\"$u \\to 0$\" loading=\"lazy\"&gt;&lt;/span&gt; por valores negativos, então &lt;!-- MATH\n ${\\mathrm{senh}}w$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1045.svg\" alt=\"${\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; vai para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img1011.svg\" alt=\"$-\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; e consequentemente, &lt;!-- MATH\n $w\n\\to -\\infty$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img1046.svg\" alt=\"$w\n\\to -\\infty$\" loading=\"lazy\"&gt;&lt;/span&gt; também. Resumindo,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P250\" title=\"2P250\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.76ex; vertical-align: -1.73ex; \" src=\"img/img1047.svg\" alt=\"$\\displaystyle \\lim_{u \\to -\\infty} {\\mathrm{csch}}^{-1} u = \\lim_{u \\to \\infty} {\\mathrm{csch}}^{-1} u = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.85ex; vertical-align: -1.82ex; \" src=\"img/img1048.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{-}} {\\mathrm{csch}}^{-1} u = -\\infty,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.85ex; vertical-align: -1.82ex; \" src=\"img/img1049.svg\" alt=\"$\\displaystyle \\lim_{u \\to 0^{+}} {\\mathrm{csch}}^{-1} u = \\infty.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P251\" title=\"2P251\"&gt;\nO gráfico desta função é representado por\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2843\"&gt;&lt;/a&gt;\n&lt;table id=\"2I27\" title=\"2I27\"&gt;\n&lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 2.27:&lt;/strong&gt;\nGráfico da função cossecante hiperbólica inversa.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;img src=\"img/farccsch.png\" alt=\"Image farccsch\" loading=\"lazy\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"2P252\" title=\"2P252\"&gt;\nSão válidas as relações de inversão,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"2P253\" title=\"2P253\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1050.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}({\\mathrm{csch}}^{-1} u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 1.86ex; vertical-align: -0.18ex; \" src=\"img/img1051.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}^{*},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1052.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}^{-1}({\\mathrm{csch}}u) = u$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 1.86ex; vertical-align: -0.18ex; \" src=\"img/img1053.svg\" alt=\"$\\displaystyle \\quad u \\in \\mathbb{R}^{*}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"2P254\" title=\"2P254\"&gt;\nA relação completa de funções trigonométricas hiperbólicas inversas com seus respectivos domínios de definição e\nconjunto imagem é dada na próxima tabela.\n\n&lt;/p&gt;\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"2868\"&gt;&lt;/a&gt;\n&lt;table id=\"2T3\" title=\"2T3\"&gt;\n&lt;caption&gt;&lt;strong&gt;Tabela 2.3:&lt;/strong&gt;\nDomínio e imagem das funções trigonométricas hiperbólicas inversas.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;table class=\"PAD  BORDER\"&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;imagem&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n ${\\mathrm{senh}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img986.svg\" alt=\"${\\mathrm{senh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n $\\cosh^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1054.svg\" alt=\"$\\cosh^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $[1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $[0,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img996.svg\" alt=\"$[0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n ${\\mathrm {tgh}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1055.svg\" alt=\"${\\mathrm {tgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n ${\\mathrm{ctgh}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1056.svg\" alt=\"${\\mathrm{ctgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n $(-\\infty,-1) \\cup (1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\mathbb{R}^{*}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n ${\\mathrm{sech}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1057.svg\" alt=\"${\\mathrm{sech}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img887.svg\" alt=\"$(0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $[0,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img996.svg\" alt=\"$[0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n ${\\mathrm{csch}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1058.svg\" alt=\"${\\mathrm{csch}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\mathbb{R}^{*}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\mathbb{R}^{*}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;",
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    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.7 Continuidade das funções trigonométricas hiperbólicas inversas",
    "text": "2.7 Continuidade das funções trigonométricas hiperbólicas inversas\n\n    &lt;p class=\" unidade\" id=\"2P255\" title=\"2P255\"&gt;\n    O procedimento adotado aqui não tem diferenças do procedimento adotado para as funções trigonométricas circulares. O\n    teorema &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teoinvcont\"&gt;1.11&lt;/a&gt; se aplica às funções trigonométricas hiperbólicas em seus respectivos domínios de definição.\n    Vamos omitir os detalhes. Entretanto entendemos deste ponto em diante que cada função trigonométrica inversa é contínua\n    nos seus respectivos domínios de definição respeitando a lateralidade nos extremos fechados destes domínios.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"2P256\" title=\"2P256\"&gt;\n    Temos assim, que\n    &lt;/p&gt;\n    &lt;div class=\"CENTER\"&gt;\n    &lt;table class=\"PAD \"&gt;\n    &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\lim\\limits_{u \\to a} {\\mathrm{senh}}^{-1} u = {\\mathrm{senh}}^{-1} a$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.61ex; vertical-align: -1.60ex; \" src=\"img/img1059.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{senh}}^{-1} u = {\\mathrm{senh}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $a \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\lim\\limits_{u \\to a} \\cosh^{-1} u = \\cosh^{-1} a$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.61ex; vertical-align: -1.60ex; \" src=\"img/img1060.svg\" alt=\"$\\lim\\limits_{u \\to a} \\cosh^{-1} u = \\cosh^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $a \\in [1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1061.svg\" alt=\"$a \\in [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\lim\\limits_{u \\to a} {\\mathrm {tgh}}^{-1} u = {\\mathrm {tgh}}^{-1} a$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.61ex; vertical-align: -1.60ex; \" src=\"img/img1062.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm {tgh}}^{-1} u = {\\mathrm {tgh}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $a \\in (-1,1)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1063.svg\" alt=\"$a \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\lim\\limits_{u \\to a} {\\mathrm{ctgh}}^{-1} u = {\\mathrm{ctgh}}^{-1} a$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.61ex; vertical-align: -1.60ex; \" src=\"img/img1064.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{ctgh}}^{-1} u = {\\mathrm{ctgh}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $a \\in (-\\infty,-1) \\cup (1,\\infty)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1065.svg\" alt=\"$a \\in (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\lim\\limits_{u \\to a} {\\mathrm{sech}}^{-1} u = {\\mathrm{sech}}^{-1} a$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.61ex; vertical-align: -1.60ex; \" src=\"img/img1066.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{sech}}^{-1} u = {\\mathrm{sech}}^{-1} a$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $a \\in (0,1]$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1067.svg\" alt=\"$a \\in (0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;,&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $\\lim\\limits_{u \\to a} {\\mathrm{csch}}^{-1} u = {\\mathrm{csch}}^{-1} u$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.61ex; vertical-align: -1.60ex; \" src=\"img/img1068.svg\" alt=\"$\\lim\\limits_{u \\to a} {\\mathrm{csch}}^{-1} u = {\\mathrm{csch}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"CENTER\"&gt;para todo&lt;/td&gt;\n    &lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n     $a \\in \\mathbb{R}^{*}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.75ex; vertical-align: -0.18ex; \" src=\"img/img1069.svg\" alt=\"$a \\in \\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;.&lt;/td&gt;\n    &lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;\n    &lt;/div&gt;",
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    "title": "Capítulo 2: Funções trigonométricas hiperbólicas",
    "section": "2.8 Derivadas das funções trigonométricas hiperbólicas inversas",
    "text": "2.8 Derivadas das funções trigonométricas hiperbólicas inversas\n\n   &lt;a id=\"secderhipinv\"&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P257\" title=\"2P257\"&gt;\n        Nesta seção, vamos determinar as fórmulas de derivada para as funções trigonométricas hiperbólicas inversas. Usaremos\n        principalmente a técnica da diferenciação implícita e levamos em conta o conhecimento das fórmulas de diferenciação\n        para as seis funções trigonométricas hiperbólicas obtidas na seção &lt;a href=\"#secderhip\"&gt;2.5&lt;/a&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P258\" title=\"2P258\"&gt;\n        Considerando a função &lt;!-- MATH\n         $w = f(u) = {\\mathrm{senh}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img1070.svg\" alt=\"$w = f(u) = {\\mathrm{senh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n         $u \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, queremos agora derivar em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e obter\n        &lt;!-- MATH\n         $w' = \\frac{dw}{du}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img556.svg\" alt=\"$w' = \\frac{dw}{du}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Sabemos que neste caso é válida a relação &lt;!-- MATH\n         $u = {\\mathrm{senh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img984.svg\" alt=\"$u = {\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembre-se que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; é variável dependente\n        de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e, por isto, quando derivarmos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; devemos usar diferenciação implícita. Nestes termos, derivando em relação a\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; os dois membros de &lt;!-- MATH\n         $u = {\\mathrm{senh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img984.svg\" alt=\"$u = {\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 = \\frac{d}{du} {\\mathrm{senh}}w = \\cosh w \\cdot \\frac{dw}{du}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P259\" title=\"2P259\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1071.svg\" alt=\"$\\displaystyle 1 = \\frac{d}{du} {\\mathrm{senh}}w = \\cosh w \\cdot \\frac{dw}{du}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P260\" title=\"2P260\"&gt;\n        Como queremos determinar &lt;!-- MATH\n         $w' = \\frac{dw}{du}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img556.svg\" alt=\"$w' = \\frac{dw}{du}$\" loading=\"lazy\"&gt;&lt;/span&gt; basta agora isolar este termo. Obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{dw}{du} = \\frac{1}{\\cosh w}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P261\" title=\"2P261\"&gt;\n        &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1072.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{1}{\\cosh w}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P262\" title=\"2P262\"&gt;\n        Mas claro que desejamos obter esta derivada como função de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; novamente. Precisamos então substituir a variável\n        dependente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img434.svg\" alt=\"$w$\" loading=\"lazy\"&gt;&lt;/span&gt; do segundo membro pela variável independente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. A única expressão que faz esta substituição é a\n        própria relação &lt;!-- MATH\n         $u = {\\mathrm{senh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img984.svg\" alt=\"$u = {\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim, vamos substituir o termo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1036.svg\" alt=\"$\\cosh\n        w$\" loading=\"lazy\"&gt;&lt;/span&gt; por alguma expressão que contenha &lt;!-- MATH\n         ${\\mathrm{senh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1045.svg\" alt=\"${\\mathrm{senh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        Usando a relação fundamental (&lt;a href=\"#idfundhip\"&gt;2.3&lt;/a&gt;), temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{du} {\\mathrm{senh}}^{-1} u = \\frac{dw}{du} = \\frac{1}{\\cosh w} = \\frac{1}{\\sqrt{ 1 + {\\mathrm{senh}}^{2} w}} = \\frac{1}{\\sqrt{1 + u^{2}}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P263\" title=\"2P263\"&gt;\n        &lt;img style=\"height: 5.45ex; vertical-align: -2.48ex; \" src=\"img/img1073.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{senh}}^{-1} u = \\frac{dw}{du} = \\frac{1}{\\c...\n        ...w} = \\frac{1}{\\sqrt{ 1 + {\\mathrm{senh}}^{2} w}} = \\frac{1}{\\sqrt{1 + u^{2}}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P264\" style=\"text-indent: 0 !important;\" title=\"2P264\"&gt;\n        para todo &lt;!-- MATH\n         $u \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P265\" title=\"2P265\"&gt;\n        Tomamos agora a função &lt;!-- MATH\n         $w = f(u) = \\cosh^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img1074.svg\" alt=\"$w = f(u) = \\cosh^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida para todo &lt;!-- MATH\n         $u \\in [1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1075.svg\" alt=\"$u \\in [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Derivando implicitamente a\n        igualdade &lt;!-- MATH\n         $u = \\cosh w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1000.svg\" alt=\"$u = \\cosh w$\" loading=\"lazy\"&gt;&lt;/span&gt; com relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1076.svg\" alt=\"$u &gt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 = ({\\mathrm{senh}}w) \\cdot \\frac{dw}{du}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P266\" title=\"2P266\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1077.svg\" alt=\"$\\displaystyle 1 = ({\\mathrm{senh}}w) \\cdot \\frac{dw}{du}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P267\" title=\"2P267\"&gt;\n        Isolando agora o termo &lt;!-- MATH\n         $\\frac{dw}{du}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1078.svg\" alt=\"$\\frac{dw}{du}$\" loading=\"lazy\"&gt;&lt;/span&gt;, como feito para o caso do seno hiperbólico e usando a relação fundamental\n        (&lt;a href=\"#idfundhip\"&gt;2.3&lt;/a&gt;), obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{du} \\cosh^{-1} u = \\frac{dw}{du} = \\frac{1}{{\\mathrm{senh}}w} = \\frac{1}{\\sqrt{ \\cosh^{2} w - 1}} = \\frac{1}{\\sqrt{u^{2} - 1}}\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P268\" title=\"2P268\"&gt;\n        &lt;img style=\"height: 5.45ex; vertical-align: -2.48ex; \" src=\"img/img1079.svg\" alt=\"$\\displaystyle \\frac{d}{du} \\cosh^{-1} u = \\frac{dw}{du} = \\frac{1}{{\\mathrm{senh}}w} = \\frac{1}{\\sqrt{ \\cosh^{2} w - 1}} = \\frac{1}{\\sqrt{u^{2} - 1}} $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P269\" style=\"text-indent: 0 !important;\" title=\"2P269\"&gt;\n        para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1076.svg\" alt=\"$u &gt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Note que esta derivada não está definida para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img309.svg\" alt=\"$u=1$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P270\" title=\"2P270\"&gt;\n        Para a função &lt;!-- MATH\n         $w = f(u) = {\\mathrm {tgh}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img1080.svg\" alt=\"$w = f(u) = {\\mathrm {tgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;, derivamos a igualdade &lt;!-- MATH\n         $u = {\\mathrm {tgh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1009.svg\" alt=\"$u = {\\mathrm {tgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; com relação\n        a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtendo\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 = ({\\mathrm{sech}}^{2} w) \\cdot \\frac{dw}{du}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P271\" title=\"2P271\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1081.svg\" alt=\"$\\displaystyle 1 = ({\\mathrm{sech}}^{2} w) \\cdot \\frac{dw}{du}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P272\" title=\"2P272\"&gt;\n        Reorganizando os termos e usando a igualdade (&lt;a href=\"#idtghsech\"&gt;2.8&lt;/a&gt;), da proposição &lt;a href=\"#propidsectghip\"&gt;2.3&lt;/a&gt;, vem\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{du} {\\mathrm {tgh}}^{-1} u = \\frac{dw}{du} = \\frac{1}{{\\mathrm{sech}}^{2} w} = \\frac{1}{1-{\\mathrm {tgh}}^{2}w} = \\frac{1}{1 - u^{2}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P273\" title=\"2P273\"&gt;\n        &lt;img style=\"height: 5.24ex; vertical-align: -2.27ex; \" src=\"img/img1082.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm {tgh}}^{-1} u = \\frac{dw}{du} = \\frac{1}{{\\mathrm{sech}}^{2} w} = \\frac{1}{1-{\\mathrm {tgh}}^{2}w} = \\frac{1}{1 - u^{2}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P274\" style=\"text-indent: 0 !important;\" title=\"2P274\"&gt;\n        para todo &lt;!-- MATH\n         $u \\in (-1,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img552.svg\" alt=\"$u \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P275\" title=\"2P275\"&gt;\n        Considerando &lt;!-- MATH\n         $w = f(u) = {\\mathrm{ctgh}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img1083.svg\" alt=\"$w = f(u) = {\\mathrm{ctgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida para todo &lt;!-- MATH\n         $u \\in (-\\infty,-1) \\cup (1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img583.svg\" alt=\"$u \\in\n        (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;, vamos derivar a\n        igualdade &lt;!-- MATH\n         $u = {\\mathrm{ctgh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1020.svg\" alt=\"$u = {\\mathrm{ctgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; com respeito a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 = (-{\\mathrm{csch}}^{2} w) \\cdot \\frac{dw}{du}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P276\" title=\"2P276\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1084.svg\" alt=\"$\\displaystyle 1 = (-{\\mathrm{csch}}^{2} w) \\cdot \\frac{dw}{du}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P277\" title=\"2P277\"&gt;\n        Isolando o termo &lt;!-- MATH\n         $w' = \\frac{dw}{du}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img556.svg\" alt=\"$w' = \\frac{dw}{du}$\" loading=\"lazy\"&gt;&lt;/span&gt; e usando a identidade (&lt;a href=\"#idctghcsch\"&gt;2.9&lt;/a&gt;) da proposição &lt;a href=\"#propidsectghip\"&gt;2.3&lt;/a&gt;,\n        temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{du} {\\mathrm{ctgh}}^{-1} u = \\frac{dw}{du} = \\frac{-1}{{\\mathrm{csch}}^{2} w} = \\frac{-1}{{\\mathrm{ctgh}}^{2}w - 1} = \\frac{-1}{u^{2}-1} = \\frac{1}{1 - u^{2}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P278\" title=\"2P278\"&gt;\n        &lt;img style=\"height: 5.24ex; vertical-align: -2.27ex; \" src=\"img/img1085.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{ctgh}}^{-1} u = \\frac{dw}{du} = \\frac{-1}{{...\n        ...frac{-1}{{\\mathrm{ctgh}}^{2}w - 1} = \\frac{-1}{u^{2}-1} = \\frac{1}{1 - u^{2}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P279\" style=\"text-indent: 0 !important;\" title=\"2P279\"&gt;\n        para &lt;!-- MATH\n         $u \\in (-\\infty,-1) \\cup (1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img583.svg\" alt=\"$u \\in\n        (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P280\" title=\"2P280\"&gt;\n        Tomando agora a função &lt;!-- MATH\n         $w = f(u) = {\\mathrm{sech}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img1086.svg\" alt=\"$w = f(u) = {\\mathrm{sech}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, que está definida para todo &lt;!-- MATH\n         $u \\in (0,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1087.svg\" alt=\"$u \\in (0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos &lt;!-- MATH\n         $u = {\\mathrm{sech}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1034.svg\" alt=\"$u = {\\mathrm{sech}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;, com\n        &lt;!-- MATH\n         $w \\in [0,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1088.svg\" alt=\"$w \\in [0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Derivando em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 = -({\\mathrm{sech}}w {\\mathrm {tgh}}w) \\cdot \\frac{dw}{du},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P281\" title=\"2P281\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1089.svg\" alt=\"$\\displaystyle 1 = -({\\mathrm{sech}}w {\\mathrm {tgh}}w) \\cdot \\frac{dw}{du}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P282\" style=\"text-indent: 0 !important;\" title=\"2P282\"&gt;\n        para todo &lt;!-- MATH\n         $u \\in (0,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img906.svg\" alt=\"$u \\in (0,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{dw}{du} = \\frac{-1}{{\\mathrm{sech}}w {\\mathrm {tgh}}w}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P283\" title=\"2P283\"&gt;\n        &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img1090.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{-1}{{\\mathrm{sech}}w {\\mathrm {tgh}}w}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P284\" title=\"2P284\"&gt;\n        Usaremos a identidade (&lt;a href=\"#idtghsech\"&gt;2.8&lt;/a&gt;) da proposição &lt;a href=\"#propidsectghip\"&gt;2.3&lt;/a&gt;, válida para &lt;!-- MATH\n         $w \\in [0,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1088.svg\" alt=\"$w \\in [0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Extraindo a\n        raiz quadrada em ambos os membros de (&lt;a href=\"#idtghsech\"&gt;2.8&lt;/a&gt;), temos que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        |{\\mathrm {tgh}}w| = \\sqrt{1-{\\mathrm{sech}}^{2} w}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P285\" title=\"2P285\"&gt;\n        &lt;img style=\"height: 3.13ex; vertical-align: -0.62ex; \" src=\"img/img1091.svg\" alt=\"$\\displaystyle \\vert{\\mathrm {tgh}}w\\vert = \\sqrt{1-{\\mathrm{sech}}^{2} w}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P286\" title=\"2P286\"&gt;\n        Como &lt;!-- MATH\n         $w \\in (0,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1092.svg\" alt=\"$w \\in (0,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; o termo &lt;!-- MATH\n         ${\\mathrm {tgh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1093.svg\" alt=\"${\\mathrm {tgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; do lado esquerdo é sempre positivo. Descartamos então o módulo, obtendo\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{du} {\\mathrm{sech}}^{-1}u = \\frac{dw}{du} = \\frac{-1}{{\\mathrm{sech}}w {\\mathrm {tgh}}w} = \\frac{-1}{{\\mathrm{sech}}w \\sqrt{1 - {\\mathrm{sech}}^{2} w}} = \\frac{-1}{u \\sqrt{1 - u^{2}}}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P287\" title=\"2P287\"&gt;\n        &lt;img style=\"height: 5.45ex; vertical-align: -2.48ex; \" src=\"img/img1094.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{sech}}^{-1}u = \\frac{dw}{du} = \\frac{-1}{{\\...\n        ...hrm{sech}}w \\sqrt{1 - {\\mathrm{sech}}^{2} w}} = \\frac{-1}{u \\sqrt{1 - u^{2}}}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P288\" title=\"2P288\"&gt;\n        Finalmente para a função &lt;!-- MATH\n         $w = f(u) = {\\mathrm{csch}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img1095.svg\" alt=\"$w = f(u) = {\\mathrm{csch}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida para todo &lt;!-- MATH\n         $u \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img824.svg\" alt=\"$u \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;, escrevemos &lt;!-- MATH\n         $u = {\\mathrm{csch}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1096.svg\" alt=\"$u = {\\mathrm{csch}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;, com\n        &lt;!-- MATH\n         $w \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1097.svg\" alt=\"$w \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; e derivando implicitamente em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 = - ({\\mathrm{csch}}w {\\mathrm{ctgh}}w) \\cdot \\frac{dw}{du},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P289\" title=\"2P289\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1098.svg\" alt=\"$\\displaystyle 1 = - ({\\mathrm{csch}}w {\\mathrm{ctgh}}w) \\cdot \\frac{dw}{du}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P290\" style=\"text-indent: 0 !important;\" title=\"2P290\"&gt;\n        que nos fornece\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{dw}{du} = \\frac{-1}{{\\mathrm{csch}}w {\\mathrm{ctgh}}w}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P291\" title=\"2P291\"&gt;\n        &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img1099.svg\" alt=\"$\\displaystyle \\frac{dw}{du} = \\frac{-1}{{\\mathrm{csch}}w {\\mathrm{ctgh}}w}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P292\" title=\"2P292\"&gt;\n        Vamos usar a igualdade (&lt;a href=\"#idctghcsch\"&gt;2.9&lt;/a&gt;), da proposição &lt;a href=\"#propidsectghip\"&gt;2.3&lt;/a&gt;, válida para &lt;!-- MATH\n         $w \\in \\mathbb{R}^{*}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.75ex; vertical-align: -0.18ex; \" src=\"img/img1100.svg\" alt=\"$w \\in \\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Extraímos a\n        raiz quadrada em ambos os membros de (&lt;a href=\"#idctghcsch\"&gt;2.9&lt;/a&gt;) para obter\n        &lt;!-- MATH\n         \\begin{displaymath}\n        |{\\mathrm{ctgh}}w| = \\sqrt{1 + {\\mathrm{csch}}^{2} w}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P293\" title=\"2P293\"&gt;\n        &lt;img style=\"height: 3.13ex; vertical-align: -0.62ex; \" src=\"img/img1101.svg\" alt=\"$\\displaystyle \\vert{\\mathrm{ctgh}}w\\vert = \\sqrt{1 + {\\mathrm{csch}}^{2} w}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P294\" title=\"2P294\"&gt;\n        Observe que &lt;!-- MATH\n         ${\\mathrm{ctgh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1102.svg\" alt=\"${\\mathrm{ctgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; não é sempre positiva para &lt;!-- MATH\n         $w \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1097.svg\" alt=\"$w \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; e isto nos impede de descartar o módulo. Mas &lt;!-- MATH\n         ${\\mathrm{csch}}\n        w {\\mathrm{ctgh}}w$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1103.svg\" alt=\"${\\mathrm{csch}}\n        w {\\mathrm{ctgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt; é sempre positivo. Então temos\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P295\" title=\"2P295\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1104.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}w {\\mathrm{ctgh}}w$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1105.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm{csch}}w {\\mathrm{ctgh}}w\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1106.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm{csch}}w\\vert \\vert{\\mathrm{ctgh}}w\\vert$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.13ex; vertical-align: -0.62ex; \" src=\"img/img1107.svg\" alt=\"$\\displaystyle = \\vert{\\mathrm{csch}}w\\vert\\sqrt{1 + {\\mathrm{csch}}^{2} w} = \\vert u\\vert \\sqrt{1+u^{2}},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"2P296\" style=\"text-indent: 0 !important;\" title=\"2P296\"&gt;\n        donde segue que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{du} {\\mathrm{csch}}^{-1} u = \\frac{dw}{du} = \\frac{-1}{{\\mathrm{csch}}w {\\mathrm{ctgh}}w} = \\frac{-1}{|u| \\sqrt{1+u^{2}}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"2P297\" title=\"2P297\"&gt;\n        &lt;img style=\"height: 5.35ex; vertical-align: -2.38ex; \" src=\"img/img1108.svg\" alt=\"$\\displaystyle \\frac{d}{du} {\\mathrm{csch}}^{-1} u = \\frac{dw}{du} = \\frac{-1}{{\\mathrm{csch}}w {\\mathrm{ctgh}}w} = \\frac{-1}{\\vert u\\vert \\sqrt{1+u^{2}}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"2P298\" style=\"text-indent: 0 !important;\" title=\"2P298\"&gt;\n        para todo &lt;!-- MATH\n         $u \\in \\mathbb{R}^{*}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.75ex; vertical-align: -0.18ex; \" src=\"img/img969.svg\" alt=\"$u \\in \\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"2P299\" title=\"2P299\"&gt;\n        Vamos resumir as fórmulas desta seção na próxima tabela.\n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"3055\"&gt;&lt;/a&gt;\n        &lt;table id=\"2T4\" title=\"2T4\"&gt;\n        &lt;caption&gt;&lt;strong&gt;Tabela 2.4:&lt;/strong&gt;\n        Derivadas das funções trigonométricas hiperbólicas inversas.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;table class=\"PAD  BORDER\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;derivada&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{senh}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img986.svg\" alt=\"${\\mathrm{senh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{1}{\\sqrt{1 + u^{2}}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img1109.svg\" alt=\"$\\frac{1}{\\sqrt{1 + u^{2}}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\cosh^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1054.svg\" alt=\"$\\cosh^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $[1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{1}{\\sqrt{u^{2} - 1}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img1110.svg\" alt=\"$\\frac{1}{\\sqrt{u^{2} - 1}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm {tgh}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1055.svg\" alt=\"${\\mathrm {tgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{1}{1 - u^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img1111.svg\" alt=\"$\\frac{1}{1 - u^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{ctgh}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1056.svg\" alt=\"${\\mathrm{ctgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $(-\\infty,-1) \\cup (1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img619.svg\" alt=\"$(-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{1}{1 - u^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img1111.svg\" alt=\"$\\frac{1}{1 - u^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{sech}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1057.svg\" alt=\"${\\mathrm{sech}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img112.svg\" alt=\"$(0,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{-1}{u \\sqrt{1 - u^{2}}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img1112.svg\" alt=\"$\\frac{-1}{u \\sqrt{1 - u^{2}}}$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{csch}}^{-1} u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1058.svg\" alt=\"${\\mathrm{csch}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\mathbb{R}^{*}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.67ex; vertical-align: -0.10ex; \" src=\"img/img880.svg\" alt=\"$\\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{-1}{|u| \\sqrt{1+u^{2}}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.54ex; vertical-align: -1.56ex; \" src=\"img/img1113.svg\" alt=\"$\\frac{-1}{\\vert u\\vert \\sqrt{1+u^{2}}}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n         &lt;a id=\"tabderhipinv\"&gt;&lt;/a&gt;\n        &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        &lt;br&gt;\n        \n        &lt;p class=\" unidade\" id=\"2P300\" title=\"2P300\"&gt;\n        Note que as derivadas das funções &lt;!-- MATH\n         ${\\mathrm {tgh}}^{-1}u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1055.svg\" alt=\"${\\mathrm {tgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         ${\\mathrm{ctgh}}^{-1}u$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1056.svg\" alt=\"${\\mathrm{ctgh}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt; são iguais, porém estão definidas em conjuntos\n        disjuntos, isto é, conjuntos que não possuem pontos em comum.\n        &lt;/p&gt;",
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      "Capítulo 2: Funções trigonométricas hiperbólicas"
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    "title": "Capítulo 3: Igualdades exponenciais e logarítmicas",
    "section": "3.1 Método das séries de potência",
    "text": "3.1 Método das séries de potência\n\n&lt;p class=\" unidade\" id=\"3P6\" title=\"3P6\"&gt;\nO método que utilizaremos para provar as identidades em (&lt;a href=\"#idexpo\"&gt;3.1&lt;/a&gt;) nesta seção é o método das séries de potência e\nentão faremos primeiramente uma breve introdução às séries de potência de variáveis reais. Para um estudo mais\naprofundado sobre séries de potências recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#Swokowski\"&gt;8&lt;/a&gt;, Swokowski].\n\n&lt;/p&gt;\n&lt;div&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;&lt;/b&gt; &nbsp; \nSe &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma variável real independente, então uma série de potências em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma soma infinita da forma,\n&lt;!-- MATH\n \\begin{displaymath}\n\\sum_{n=0}^{\\infty} a_{n} x^{n} = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} + a_{5}x^{5} + \\cdots\n\\end{displaymath}\n --&gt;\n\n&lt;div class=\"mathdisplay unidade\" id=\"3P7\" title=\"3P7\"&gt;\n&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1116.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} a_{n} x^{n} = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} + a_{5}x^{5} + \\cdots $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\nsendo que cada termo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt; é um número real, chamado de coeficiente da &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt;-ésima potência de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P8\" title=\"3P8\"&gt;\nObserve que o primeiro termo da série é &lt;!-- MATH\n $a_{0}x^{0}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.46ex; \" src=\"img/img1119.svg\" alt=\"$a_{0}x^{0}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para simplificar a notação, estamos supondo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.03ex; vertical-align: -0.11ex; \" src=\"img/img1120.svg\" alt=\"$x^{0} = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; mesmo\npara &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1121.svg\" alt=\"$x=0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P9\" title=\"3P9\"&gt;\nSe a soma infinita existir e for um número real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img7.svg\" alt=\"$S$\" loading=\"lazy\"&gt;&lt;/span&gt;, então dizemos que a série converge, ou ainda, que converge para\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img7.svg\" alt=\"$S$\" loading=\"lazy\"&gt;&lt;/span&gt;. Se a soma não existir então a série é dita divergente. Naturalmente a convergência de uma série de potências está\ncondicionada aos termos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt; e principalmente ao valor da variável &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P10\" title=\"3P10\"&gt;\nObserve que para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1121.svg\" alt=\"$x=0$\" loading=\"lazy\"&gt;&lt;/span&gt; a série se reduz a um único termo e, portanto, é uma série convergente (para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1122.svg\" alt=\"$a_{0}$\" loading=\"lazy\"&gt;&lt;/span&gt;). O que\nrealmente interessa é se existem outros valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;, além de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1121.svg\" alt=\"$x=0$\" loading=\"lazy\"&gt;&lt;/span&gt;, para os quais a série de potências é\nconvergente. Nestes termos um fato importante é a determinação dos valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; que tornam uma série de potências\nconvergente, isto é, determinar os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; para os quais a soma infinita existe.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P11\" title=\"3P11\"&gt;\nO conjunto dos valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; que tornam a série convergente é um intervalo, centrado em &lt;span class=\"MATH\"&gt;0&lt;/span&gt; e com raio &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1123.svg\" alt=\"$r&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;. É um\nintervalo do tipo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1124.svg\" alt=\"$(-r,r)$\" loading=\"lazy\"&gt;&lt;/span&gt;, podendo ainda ser fechado em algum dos extremos. Para determinar este intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1124.svg\" alt=\"$(-r,r)$\" loading=\"lazy\"&gt;&lt;/span&gt;\nusamos, em geral, o chamado teste da razão (Critério de D'Alembert).\n&lt;/p&gt;\n&lt;div id=\"3Teo2\" title=\"3Teo2\" class=\"unidade\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;&lt;/b&gt; (Teste da razão)  &nbsp; \n&lt;i&gt;Dada uma série &lt;!-- MATH\n $\\sum b_{n}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1125.svg\" alt=\"$\\sum b_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então\n&lt;/i&gt;&lt;table width=\"90%\"&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1126.svg\" alt=\"$i)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Se &lt;!-- MATH\n $\\lim\\limits_{n \\to \\infty} \\left| \\frac{b_{n+1}}{b_{n}} \\right| = L &lt; 1$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.12ex; vertical-align: -1.60ex; \" src=\"img/img1127.svg\" alt=\"$\\lim\\limits_{n \\to \\infty} \\left\\vert \\frac{b_{n+1}}{b_{n}} \\right\\vert = L &lt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, a série é absolutamente\nconvergente (e, portanto, convergente).\n&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;&lt;td align=\"right\" valign=\"top\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1128.svg\" alt=\"$ii)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;&lt;td valign=\"top\"&gt;&nbsp;Se &lt;!-- MATH\n $\\lim\\limits_{n \\to \\infty} \\left| \\frac{b_{n+1}}{b_{n}} \\right| = L &gt; 1$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.12ex; vertical-align: -1.60ex; \" src=\"img/img1129.svg\" alt=\"$\\lim\\limits_{n \\to \\infty} \\left\\vert \\frac{b_{n+1}}{b_{n}} \\right\\vert = L &gt; 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, a série é divergente.\n&lt;/td&gt;&lt;/tr&gt;&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P12\" title=\"3P12\"&gt;\nPara os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; que tornam a série convergente, definimos uma função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, cujo domínio é o intervalo de\nconvergência da série. O recíproco disto é uma pergunta mais interessante. Dada uma função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; definida em algum\nintervalo &lt;!-- MATH\n $I = (-r,r)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1130.svg\" alt=\"$I = (-r,r)$\" loading=\"lazy\"&gt;&lt;/span&gt;, é possível obter uma série de potências &lt;!-- MATH\n $\\sum a_{n}x^{n}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1131.svg\" alt=\"$\\sum a_{n}x^{n}$\" loading=\"lazy\"&gt;&lt;/span&gt; de forma que &lt;!-- MATH\n $f(x) = \\sum a_{n}x^{n}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1132.svg\" alt=\"$f(x) = \\sum a_{n}x^{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;\npara todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1133.svg\" alt=\"$x \\in I$\" loading=\"lazy\"&gt;&lt;/span&gt;? Mais ainda, se existir tal série, como devem ser os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;? A segunda pergunta\né respondida pelo teorema de Maclaurin.\n\n&lt;/p&gt;\n&lt;div id=\"3Teo3\" title=\"3Teo3\" class=\"unidade\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;&lt;/b&gt; (Teorema de Maclaurin)  &nbsp; \n&lt;i&gt;Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função que admite uma representação por série de potências de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n&lt;/i&gt;&lt;!-- MATH\n \\begin{displaymath}\nf(x) = \\sum_{n=0}^{\\infty} a_{n} x^{n}\n\\end{displaymath}\n --&gt;\n\n&lt;div class=\"mathdisplay unidade\" id=\"3P13\" title=\"3P13\"&gt;\n&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1134.svg\" alt=\"$\\displaystyle f(x) = \\sum_{n=0}^{\\infty} a_{n} x^{n} $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P14\" style=\"text-indent: 0 !important;\" title=\"3P14\"&gt;&lt;i&gt;\npara todo &lt;!-- MATH\n $x \\in (-r,r)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1135.svg\" alt=\"$x \\in (-r,r)$\" loading=\"lazy\"&gt;&lt;/span&gt;, então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função infinitamente diferenciável no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1121.svg\" alt=\"$x=0$\" loading=\"lazy\"&gt;&lt;/span&gt; e mais ainda,\n&lt;/i&gt;&lt;!-- MATH\n \\begin{displaymath}\nf(x) = f(0) + f'(0)x + \\frac{f''(0)}{2} x^{2} + \\frac{f'''(0)}{3!} x^{3} + \\frac{f^{(4)}(0)}{4!} x^{4} + \\cdots + \\frac{f^{(n)}(0)}{n!} x^{n} + \\cdots\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P15\" title=\"3P15\"&gt;\n&lt;img style=\"height: 5.01ex; vertical-align: -1.57ex; \" src=\"img/img1136.svg\" alt=\"$\\displaystyle f(x) = f(0) + f'(0)x + \\frac{f''(0)}{2} x^{2} + \\frac{f'''(0)}{3!...\n... + \\frac{f^{(4)}(0)}{4!} x^{4} + \\cdots + \\frac{f^{(n)}(0)}{n!} x^{n} + \\cdots $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;i&gt;\npara todo &lt;!-- MATH\n $x \\in (-r,r)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1135.svg\" alt=\"$x \\in (-r,r)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/i&gt;&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P16\" title=\"3P16\"&gt;\nEste teorema nos diz principalmente que os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;, da série de potências de uma função, são\nrespectivamente &lt;!-- MATH\n $\\frac{f^{(n)}(0)}{n!}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.36ex; vertical-align: -0.83ex; \" src=\"img/img1137.svg\" alt=\"$\\frac{f^{(n)}(0)}{n!}$\" loading=\"lazy\"&gt;&lt;/span&gt;, sendo que a notação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1138.svg\" alt=\"$f^{(n)}$\" loading=\"lazy\"&gt;&lt;/span&gt; refere-se à derivada de ordem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; da função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P17\" title=\"3P17\"&gt;\nPara exemplificar o processo, vamos obter as séries de potência de algumas funções de interesse como &lt;!-- MATH\n ${\\mathrm{senh}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1139.svg\" alt=\"${\\mathrm{senh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1140.svg\" alt=\"$\\cosh\nx$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;!-- MATH\n ${\\mathrm {sen}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1141.svg\" alt=\"${\\mathrm {sen}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1142.svg\" alt=\"$\\cos x$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1143.svg\" alt=\"$e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt;, assumindo que estas funções admitem uma representação em série de potências em algum\nintervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1124.svg\" alt=\"$(-r,r)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Este não deve ser um trabalho difícil neste momento pois conhecemos as derivadas destas funções.\nAssim, parece não haver problemas significativos para a determinação dos coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt; das séries de potências\ndestas funções.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P18\" title=\"3P18\"&gt;\nPrimeiramente, vamos à série de potências da função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1143.svg\" alt=\"$e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esperamos encontrar uma série de potências em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;, de\nforma que,\n&lt;!-- MATH\n \\begin{displaymath}\ne^{x} = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} + a_{5}x^{5} + \\cdots.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P19\" title=\"3P19\"&gt;\n&lt;img style=\"height: 2.49ex; vertical-align: -0.46ex; \" src=\"img/img1144.svg\" alt=\"$\\displaystyle e^{x} = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} + a_{5}x^{5} + \\cdots. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P20\" title=\"3P20\"&gt;\nDe acordo com o teorema de Maclaurin, devemos ter &lt;!-- MATH\n $a_{n} = \\frac{f^{(n)}(0)}{n!}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.36ex; vertical-align: -0.83ex; \" src=\"img/img1145.svg\" alt=\"$a_{n} = \\frac{f^{(n)}(0)}{n!}$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n $n \\in \\mathbb{N}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1146.svg\" alt=\"$n \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como sabemos,\na função exponencial &lt;!-- MATH\n $f(x) = e^{x}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1147.svg\" alt=\"$f(x) = e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt; possui derivadas de qualquer ordem contínuas e mais ainda &lt;!-- MATH\n $f^{(n)}(x) = e^{x}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1148.svg\" alt=\"$f^{(n)}(x) = e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt; para\nqualquer &lt;!-- MATH\n $n \\in \\mathbb{N}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1146.svg\" alt=\"$n \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Então os coeficientes da série de Maclaurin ficam\n&lt;!-- MATH\n \\begin{displaymath}\na_{n} = \\frac{f^{(n)}(0)}{n!} = \\frac{e^{0}}{n!} = \\frac{1}{n!},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P21\" title=\"3P21\"&gt;\n&lt;img style=\"height: 4.98ex; vertical-align: -1.55ex; \" src=\"img/img1149.svg\" alt=\"$\\displaystyle a_{n} = \\frac{f^{(n)}(0)}{n!} = \\frac{e^{0}}{n!} = \\frac{1}{n!}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P22\"  style=\"text-indent: 0 !important;\" title=\"3P22\"&gt;\ne assim temos que\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P23\" title=\"3P23\"&gt;&lt;a id=\"exserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n \\begin{equation}\ne^{x} = 1 + x + \\frac{1}{2}x^{2} + \\frac{1}{3!}x^{3} + \\frac{1}{4!}x^{4} + \\frac{1}{5!}x^{5} + \\frac{1}{6!}x^{6} + \\frac{1}{7!}x^{7} + \\cdots.\n\\end{equation}\n --&gt;\n&lt;table&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1150.svg\" alt=\"$\\displaystyle e^{x} = 1 + x + \\frac{1}{2}x^{2} + \\frac{1}{3!}x^{3} + \\frac{1}{4!}x^{4} + \\frac{1}{5!}x^{5} + \\frac{1}{6!}x^{6} + \\frac{1}{7!}x^{7} + \\cdots.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n    (&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"3P24\" title=\"3P24\"&gt;\nPodemos determinar (pelo teste da razão) que a série do lado direito converge para todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, já que para qualquer &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{n \\to \\infty} \\left| \\frac{\\frac{1}{(n+1)!} x^{n+1}}{ \\frac{1}{n!} x^{n}} \\right|\n = \\lim_{n \\to \\infty} \\left| \\frac{n!}{(n+1)!} \\frac{x^{n+1}}{x^{n}} \\right| = \\lim_{n \\to \\infty} \\frac{1}{n+1} |x| = 0 &lt; 1, \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P25\" title=\"3P25\"&gt;\n&lt;img style=\"height: 6.68ex; vertical-align: -2.74ex; \" src=\"img/img1152.svg\" alt=\"$\\displaystyle \\lim_{n \\to \\infty} \\left\\vert \\frac{\\frac{1}{(n+1)!} x^{n+1}}{ \\...\n...}{x^{n}} \\right\\vert = \\lim_{n \\to \\infty} \\frac{1}{n+1} \\vert x\\vert = 0 &lt; 1, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P26\"  style=\"text-indent: 0 !important;\" title=\"3P26\"&gt;\ne, portanto, a igualdade (&lt;a href=\"#exserie\"&gt;3.3&lt;/a&gt;) é válida para todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P27\" title=\"3P27\"&gt;\nConsideremos agora a função &lt;!-- MATH\n $f(x) = {\\mathrm{senh}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1153.svg\" alt=\"$f(x) = {\\mathrm{senh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Queremos determinar os coeficientes &lt;!-- MATH\n $a_{n} = \\frac{f^{(n)}(0)}{n!}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.36ex; vertical-align: -0.83ex; \" src=\"img/img1145.svg\" alt=\"$a_{n} = \\frac{f^{(n)}(0)}{n!}$\" loading=\"lazy\"&gt;&lt;/span&gt; para\ntodo &lt;!-- MATH\n $n \\in \\mathbb{N}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1146.svg\" alt=\"$n \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Dos resultados dos capítulos anteriores, temos que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é par, &lt;!-- MATH\n $f^{(n)}(x) = {\\mathrm{senh}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1154.svg\" alt=\"$f^{(n)}(x) = {\\mathrm{senh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt; e se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é\nímpar &lt;!-- MATH\n $f^{(n)}(x) = \\cosh x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1155.svg\" alt=\"$f^{(n)}(x) = \\cosh x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta forma, os coeficientes são dados por\n&lt;!-- MATH\n \\begin{displaymath}\na_{n} = \\left\\{ \\begin{array}{ll}\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{{\\mathrm{senh}}0}{n!} = \\dfrac{0}{n!} = 0 & \\text{se} \\quad n \\quad \\text{é par}, \\\\\n                   & \\\\\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{\\cosh 0}{n!} = \\dfrac{1}{n!} & \\text{se} \\quad n \\quad \\text{é ímpar}.\n                   \\end{array} \\right. \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P28\" title=\"3P28\"&gt;\n&lt;img style=\"height: 12.90ex; vertical-align: -5.94ex; \" src=\"img/img1156.svg\" alt=\"$\\displaystyle a_{n} = \\left\\{ \\begin{array}{ll}\n\\dfrac{f^{(n)}(0)}{n!} = \\dfr...\n...dfrac{1}{n!} &amp; \\text{se} \\quad n \\quad \\text{é ímpar}.\n\\end{array} \\right. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P29\" title=\"3P29\"&gt;\nSubstituindo estes coeficientes na série de potências temos que\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P30\" title=\"3P30\"&gt;&lt;a id=\"sinhxserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n \\begin{equation}\n{\\mathrm{senh}}x = x + \\frac{1}{3!}x^{3} + \\frac{1}{5!}x^{5} + \\frac{1}{7!}x^{7} + \\frac{1}{9!}x^{9} + \\frac{1}{11!}x^{11} + \\cdots.\n\\end{equation}\n --&gt;\n&lt;table &gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1157.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}x = x + \\frac{1}{3!}x^{3} + \\frac{1}{5!}x^{5} + \\frac{1}{7!}x^{7} + \\frac{1}{9!}x^{9} + \\frac{1}{11!}x^{11} + \\cdots.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n    (&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"3P31\" title=\"3P31\"&gt;\nA série do lado direito converge (pelo teste da razão) para qualquer &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, pois para todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; temos\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{n \\to \\infty} \\left| \\frac{\\frac{1}{(2n+1)!} x^{2n+1}}{ \\frac{1}{(2n-1)!} x^{2n-1}} \\right|\n = \\lim_{n \\to \\infty} \\left| \\frac{(2n-1)!}{(2n+1)!} \\frac{x^{2n+1}}{x^{2n-1}} \\right| = \\lim_{n \\to \\infty} \\frac{1}{(2n+1)2n} |x^{2}| = 0 &lt; 1, \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P32\" title=\"3P32\"&gt;\n&lt;img style=\"height: 6.84ex; vertical-align: -2.91ex; \" src=\"img/img1158.svg\" alt=\"$\\displaystyle \\lim_{n \\to \\infty} \\left\\vert \\frac{\\frac{1}{(2n+1)!} x^{2n+1}}{...\n...\\right\\vert = \\lim_{n \\to \\infty} \\frac{1}{(2n+1)2n} \\vert x^{2}\\vert = 0 &lt; 1, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P33\"  style=\"text-indent: 0 !important;\" title=\"3P33\"&gt;\ne, portanto, a igualdade (&lt;a href=\"#sinhxserie\"&gt;3.4&lt;/a&gt;) é válida para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; real.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P34\" title=\"3P34\"&gt;\nCom raciocínio similar desenvolvemos a série de potências para a função &lt;!-- MATH\n $f(x) = \\cosh x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1159.svg\" alt=\"$f(x) = \\cosh x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembremos que agora,\n&lt;!-- MATH\n $f^{(n)}(x) = \\cosh x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1155.svg\" alt=\"$f^{(n)}(x) = \\cosh x$\" loading=\"lazy\"&gt;&lt;/span&gt; se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é par e &lt;!-- MATH\n $f^{(n)}(x) = {\\mathrm{senh}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1154.svg\" alt=\"$f^{(n)}(x) = {\\mathrm{senh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt; se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é ímpar. Então, contrariamente ao caso anterior,\n&lt;!-- MATH\n \\begin{displaymath}\na_{n} = \\left\\{ \\begin{array}{ll}\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{\\cosh 0}{n!} = \\dfrac{1}{n!} & \\text{se} \\quad n \\quad \\text{é par}, \\\\\n                   & \\\\\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{{\\mathrm{senh}}0}{n!} = \\dfrac{0}{n!} = 0 & \\text{se} \\quad n \\quad \\text{é ímpar}.\n                   \\end{array} \\right. \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P35\" title=\"3P35\"&gt;\n&lt;img style=\"height: 12.90ex; vertical-align: -5.94ex; \" src=\"img/img1160.svg\" alt=\"$\\displaystyle a_{n} = \\left\\{ \\begin{array}{ll}\n\\dfrac{f^{(n)}(0)}{n!} = \\dfr...\n...c{0}{n!} = 0 &amp; \\text{se} \\quad n \\quad \\text{é ímpar}.\n\\end{array} \\right. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P36\" title=\"3P36\"&gt;\nDesta forma, temos\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P37\" title=\"3P37\"&gt;&lt;a id=\"coshxserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n \\begin{equation}\n\\cosh x = 1 + \\frac{1}{2!}x^{2} + \\frac{1}{4!}x^{4} + \\frac{1}{6!}x^{6} + \\frac{1}{8!}x^{8} + \\frac{1}{10!}x^{10} + \\cdots,\n\\end{equation}\n --&gt;\n&lt;table&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1161.svg\" alt=\"$\\displaystyle \\cosh x = 1 + \\frac{1}{2!}x^{2} + \\frac{1}{4!}x^{4} + \\frac{1}{6!}x^{6} + \\frac{1}{8!}x^{8} + \\frac{1}{10!}x^{10} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n    (&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;)  &lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;p class=\" unidade\" id=\"3P38\"  style=\"text-indent: 0 !important;\" title=\"3P38\"&gt;\nsendo também esta igualdade verdadeira para todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P39\" title=\"3P39\"&gt;\nDada agora a função &lt;!-- MATH\n $f(x) = {\\mathrm {sen}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1162.svg\" alt=\"$f(x) = {\\mathrm {sen}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;, queremos determinar para todo &lt;!-- MATH\n $n \\in \\mathbb{N}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1146.svg\" alt=\"$n \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt; os coeficientes &lt;!-- MATH\n $a_{n} =\n\\frac{f^{(n)}(0)}{n!}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.36ex; vertical-align: -0.83ex; \" src=\"img/img1145.svg\" alt=\"$a_{n} = \\frac{f^{(n)}(0)}{n!}$\" loading=\"lazy\"&gt;&lt;/span&gt; da série de potências de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Dos resultados anteriores, sabemos que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é par,\n&lt;!-- MATH\n $f^{(n)}(x) = (-1)^{\\frac{n}{2}} {\\mathrm {sen}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.71ex; vertical-align: -0.62ex; \" src=\"img/img1163.svg\" alt=\"$f^{(n)}(x) = (-1)^{\\frac{n}{2}} {\\mathrm {sen}}x$\" loading=\"lazy\"&gt;&lt;/span&gt; e se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é ímpar &lt;!-- MATH\n $f^{(n)}(x) = (-1)^{\\frac{n-1}{2}} \\cos x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.98ex; vertical-align: -0.62ex; \" src=\"img/img1164.svg\" alt=\"$f^{(n)}(x) = (-1)^{\\frac{n-1}{2}} \\cos x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Desta forma, os\ncoeficientes são\n&lt;!-- MATH\n \\begin{displaymath}\na_{n} = \\left\\{ \\begin{array}{ll}\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{(-1)^{\\frac{n}{2}} {\\mathrm {sen}}0}{n!} = \\dfrac{0}{n!} = 0 & \\text{se} \\quad n \\quad \\text{é par}, \\\\\n                   & \\\\\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{(-1)^{\\frac{n-1}{2}} \\cos 0}{n!} = \\dfrac{(-1)^{\\frac{n-1}{2}}}{n!} & \\text{se} \\quad n \\quad \\text{é ímpar}.\n                   \\end{array} \\right. \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P40\" title=\"3P40\"&gt;\n&lt;img style=\"height: 13.31ex; vertical-align: -6.15ex; \" src=\"img/img1165.svg\" alt=\"$\\displaystyle a_{n} = \\left\\{ \\begin{array}{ll}\n\\dfrac{f^{(n)}(0)}{n!} = \\dfr...\n...-1}{2}}}{n!} &amp; \\text{se} \\quad n \\quad \\text{é ímpar}.\n\\end{array} \\right. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P41\" title=\"3P41\"&gt;\nObserve que os termos &lt;!-- MATH\n $(-1)^{\\frac{n-1}{2}}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.98ex; vertical-align: -0.62ex; \" src=\"img/img1166.svg\" alt=\"$(-1)^{\\frac{n-1}{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; ímpar, são alternadamente 1 e -1. Substituindo estes coeficientes\nna série de potências temos que\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P42\" title=\"3P42\"&gt;&lt;a id=\"sinxserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n \\begin{equation}\n{\\mathrm {sen}}x = x - \\frac{1}{3!}x^{3} + \\frac{1}{5!}x^{5} - \\frac{1}{7!}x^{7} + \\frac{1}{9!}x^{9} - \\frac{1}{11!}x^{11} + \\cdots.\n\\end{equation}\n --&gt;\n&lt;table&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1167.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}x = x - \\frac{1}{3!}x^{3} + \\frac{1}{5!}x^{5} - \\frac{1}{7!}x^{7} + \\frac{1}{9!}x^{9} - \\frac{1}{11!}x^{11} + \\cdots.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n    (&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;)   &lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"3P43\" title=\"3P43\"&gt;\nA série do lado direito converge (pelo teste da razão) para qualquer &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, já que\n&lt;!-- MATH\n \\begin{displaymath}\n\\lim_{n \\to \\infty} \\left|- \\frac{\\frac{1}{(2n+1)!} x^{2n+1}}{ \\frac{1}{(2n-1)!} x^{2n-1}} \\right|\n = \\lim_{n \\to \\infty} \\left| \\frac{(2n-1)!}{(2n+1)!} \\frac{x^{2n+1}}{x^{2n-1}} \\right| = \\lim_{n \\to \\infty} \\frac{1}{(2n+1)2n} |x^{2}| = 0 &lt; 1, \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P44\" title=\"3P44\"&gt;\n&lt;img style=\"height: 6.84ex; vertical-align: -2.91ex; \" src=\"img/img1168.svg\" alt=\"$\\displaystyle \\lim_{n \\to \\infty} \\left\\vert- \\frac{\\frac{1}{(2n+1)!} x^{2n+1}}...\n...\\right\\vert = \\lim_{n \\to \\infty} \\frac{1}{(2n+1)2n} \\vert x^{2}\\vert = 0 &lt; 1, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P45\"  style=\"text-indent: 0 !important;\" title=\"3P45\"&gt;\npara qualquer &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Segue que a igualdade (&lt;a href=\"#sinxserie\"&gt;3.6&lt;/a&gt;) é válida para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; real.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P46\" title=\"3P46\"&gt;\nCom raciocínio similar a este abordamos a função &lt;!-- MATH\n $f(x) = \\cos x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1169.svg\" alt=\"$f(x) = \\cos x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Lembremos que, &lt;!-- MATH\n $f^{(n)}(x) = (-1)^{\\frac{n}{2}} \\cos\nx$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.71ex; vertical-align: -0.62ex; \" src=\"img/img1170.svg\" alt=\"$f^{(n)}(x) = (-1)^{\\frac{n}{2}} \\cos\nx$\" loading=\"lazy\"&gt;&lt;/span&gt; se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é par e &lt;!-- MATH\n $f^{(n)}(x) = (-1)^{\\frac{n+1}{2}} {\\mathrm {sen}}x$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.98ex; vertical-align: -0.62ex; \" src=\"img/img1171.svg\" alt=\"$f^{(n)}(x) = (-1)^{\\frac{n+1}{2}} {\\mathrm {sen}}x$\" loading=\"lazy\"&gt;&lt;/span&gt; se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; é ímpar. Então, contrariamente ao caso anterior,\n&lt;!-- MATH\n \\begin{displaymath}\na_{n} = \\left\\{ \\begin{array}{ll}\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{(-1)^{\\frac{n}{2}} \\cos 0}{n!} = \\dfrac{(-1)^{\\frac{n}{2}}}{n!} & \\text{se} \\quad n \\quad \\text{é par}, \\\\\n                   & \\\\\n                   \\dfrac{f^{(n)}(0)}{n!} = \\dfrac{(-1)^{\\frac{n+1}{2}} {\\mathrm {sen}}0}{n!} = \\dfrac{0}{n!} = 0 & \\text{se} \\quad n \\quad \\text{é ímpar}.\n                   \\end{array} \\right. \n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P47\" title=\"3P47\"&gt;\n&lt;img style=\"height: 13.31ex; vertical-align: -6.15ex; \" src=\"img/img1172.svg\" alt=\"$\\displaystyle a_{n} = \\left\\{ \\begin{array}{ll}\n\\dfrac{f^{(n)}(0)}{n!} = \\dfr...\n...c{0}{n!} = 0 &amp; \\text{se} \\quad n \\quad \\text{é ímpar}.\n\\end{array} \\right. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P48\" title=\"3P48\"&gt;\nDesta forma, temos\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P49\" title=\"3P49\"&gt;&lt;a id=\"cosxserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n \\begin{equation}\n\\cos x = 1 - \\frac{1}{2!}x^{2} + \\frac{1}{4!}x^{4} - \\frac{1}{6!}x^{6} + \\frac{1}{8!}x^{8} - \\frac{1}{10!}x^{10} + \\cdots,\n\\end{equation}\n --&gt;\n&lt;table&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1173.svg\" alt=\"$\\displaystyle \\cos x = 1 - \\frac{1}{2!}x^{2} + \\frac{1}{4!}x^{4} - \\frac{1}{6!}x^{6} + \\frac{1}{8!}x^{8} - \\frac{1}{10!}x^{10} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;p class=\" unidade\" id=\"3P50\" style=\"text-indent: 0 !important;\" title=\"3P50\"&gt;\nsendo também esta igualdade verdadeira para todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P51\" title=\"3P51\"&gt;\nVamos usar agora as séries de potência obtidas anteriormente para verificar a validade das igualdades em\n(&lt;a href=\"#idexpo\"&gt;3.1&lt;/a&gt;). Consideremos primeiro a série de potência da função exponencial em &lt;a href=\"#secspot\"&gt;3.1&lt;/a&gt;,\n&lt;!-- MATH\n \\begin{displaymath}\ne^{x} = 1 + x + \\frac{1}{2}x^{2} + \\frac{1}{3!}x^{3} + \\frac{1}{4!}x^{4} + \\frac{1}{5!}x^{5} + \\frac{1}{6!}x^{6} + \\frac{1}{7!}x^{7} + \\cdots,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P52\" title=\"3P52\"&gt;\n&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1174.svg\" alt=\"$\\displaystyle e^{x} = 1 + x + \\frac{1}{2}x^{2} + \\frac{1}{3!}x^{3} + \\frac{1}{4!}x^{4} + \\frac{1}{5!}x^{5} + \\frac{1}{6!}x^{6} + \\frac{1}{7!}x^{7} + \\cdots, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P53\" title=\"3P53\"&gt;\nválida para todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Naturalmente se substituirmos na série &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.27ex; \" src=\"img/img1175.svg\" alt=\"$-x$\" loading=\"lazy\"&gt;&lt;/span&gt; obtemos a série de potências para a\nfunção &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.85ex; vertical-align: -0.10ex; \" src=\"img/img1176.svg\" alt=\"$e^{-x}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto também pode ser feito como anteriormente, determinando-se os coeficientes &lt;!-- MATH\n $a_{n} =\n\\frac{f^{(n)}(0)}{n!}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.36ex; vertical-align: -0.83ex; \" src=\"img/img1145.svg\" alt=\"$a_{n} = \\frac{f^{(n)}(0)}{n!}$\" loading=\"lazy\"&gt;&lt;/span&gt; para a série de potências da função &lt;!-- MATH\n $f(x) = e^{-x}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.62ex; \" src=\"img/img1177.svg\" alt=\"$f(x) = e^{-x}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Isto nos leva aos coeficientes &lt;!-- MATH\n $a_{n} =\n\\frac{(-1)^{n}}{n!}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.11ex; vertical-align: -0.83ex; \" src=\"img/img1178.svg\" alt=\"$a_{n} =\n\\frac{(-1)^{n}}{n!}$\" loading=\"lazy\"&gt;&lt;/span&gt;. De qualquer forma teremos\n&lt;!-- MATH\n \\begin{displaymath}\ne^{-x} = 1 - x + \\frac{1}{2}x^{2} - \\frac{1}{3!}x^{3} + \\frac{1}{4!}x^{4} - \\frac{1}{5!}x^{5} + \\frac{1}{6!}x^{6} - \\frac{1}{7!}x^{7} + \\cdots,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P54\" title=\"3P54\"&gt;\n&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1179.svg\" alt=\"$\\displaystyle e^{-x} = 1 - x + \\frac{1}{2}x^{2} - \\frac{1}{3!}x^{3} + \\frac{1}{4!}x^{4} - \\frac{1}{5!}x^{5} + \\frac{1}{6!}x^{6} - \\frac{1}{7!}x^{7} + \\cdots, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P55\" style=\"text-indent: 0 !important;\" title=\"3P55\"&gt;\npara todo &lt;!-- MATH\n $x \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Somando as série de potências de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1143.svg\" alt=\"$e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt; e de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.85ex; vertical-align: -0.10ex; \" src=\"img/img1176.svg\" alt=\"$e^{-x}$\" loading=\"lazy\"&gt;&lt;/span&gt; obtemos,\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P56\" title=\"3P56\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.27ex; \" src=\"img/img1180.svg\" alt=\"$\\displaystyle e^{x} + e^{-x}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1181.svg\" alt=\"$\\displaystyle = 2 + 2\\frac{1}{2}x^{2} + 2\\frac{1}{4!}x^{4} + 2\\frac{1}{6!}x^{6} + 2\\frac{1}{8!}x^{8} + 2\\frac{1}{10!}x^{10} + \\cdots$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img1182.svg\" alt=\"$\\displaystyle = 2 \\left( 1 + \\frac{1}{2}x^{2} + \\frac{1}{4!}x^{4} + \\frac{1}{6!}x^{6} + \\frac{1}{8!}x^{8} + \\frac{1}{10!}x^{10} + \\cdots \\right).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"3P57\" title=\"3P57\"&gt;\nAgora lembremos que a série do lado direito é exatamente a série de potências da função cosseno hiperbólico (Ver\n(&lt;a href=\"#coshxserie\"&gt;3.5&lt;/a&gt;)). Desta forma temos que\n&lt;!-- MATH\n \\begin{displaymath}\ne^{x} + e^{-x} = 2 \\cosh x,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P58\" title=\"3P58\"&gt;\n&lt;img style=\"height: 2.13ex; vertical-align: -0.27ex; \" src=\"img/img1183.svg\" alt=\"$\\displaystyle e^{x} + e^{-x} = 2 \\cosh x, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P59\" style=\"text-indent: 0 !important;\" title=\"3P59\"&gt;\ndonde segue que\n&lt;!-- MATH\n \\begin{displaymath}\n\\cosh x = \\frac{e^{x} + e^{-x}}{2}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P60\" title=\"3P60\"&gt;\n&lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1184.svg\" alt=\"$\\displaystyle \\cosh x = \\frac{e^{x} + e^{-x}}{2}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P61\" title=\"3P61\"&gt;\nPor outro lado, fazendo a diferença entre as séries de potências das funções &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1143.svg\" alt=\"$e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.85ex; vertical-align: -0.10ex; \" src=\"img/img1176.svg\" alt=\"$e^{-x}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P62\" title=\"3P62\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.27ex; \" src=\"img/img1185.svg\" alt=\"$\\displaystyle e^{x} - e^{-x}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1186.svg\" alt=\"$\\displaystyle = 2x + 2\\frac{1}{3!}x^{3} + 2\\frac{1}{5!}x^{5} + 2\\frac{1}{7!}x^{7} + 2\\frac{1}{9!}x^{9} + 2\\frac{1}{11!}x^{11} + \\cdots$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td&gt;&nbsp;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img1187.svg\" alt=\"$\\displaystyle = 2 \\left( x + \\frac{1}{3!}x^{3} + \\frac{1}{5!}x^{5} + \\frac{1}{7!}x^{7} + \\frac{1}{9!}x^{9} + \\frac{1}{11!}x^{11} + \\cdots \\right),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n&lt;p class=\" unidade\" id=\"3P63\" style=\"text-indent: 0 !important;\" title=\"3P63\"&gt;\ne lembrando que o lado direito é a série de potências da função seno hiperbólico (Ver (&lt;a href=\"#sinhxserie\"&gt;3.4&lt;/a&gt;)), temos que\n&lt;!-- MATH\n \\begin{displaymath}\ne^{x} - e^{-x} = 2 {\\mathrm{senh}}x,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P64\" title=\"3P64\"&gt;\n&lt;img style=\"height: 2.13ex; vertical-align: -0.27ex; \" src=\"img/img1188.svg\" alt=\"$\\displaystyle e^{x} - e^{-x} = 2 {\\mathrm{senh}}x, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P65\" style=\"text-indent: 0 !important;\" title=\"3P65\"&gt;\ndonde segue\n&lt;!-- MATH\n \\begin{displaymath}\n{\\mathrm{senh}}x = \\frac{e^{x} - e^{-x}}{2}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P66\" title=\"3P66\"&gt;\n&lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1189.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}x = \\frac{e^{x} - e^{-x}}{2}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P67\" title=\"3P67\"&gt;\nFica assim verificada a validade das fórmulas exponenciais, que são comumente utilizadas para definir as funções\ntrigonométricas hiperbólicas. Com estas duas igualdades, podemos escrever as demais funções trigonométricas\nhiperbólicas também em termos da função exponencial. São\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P68\" title=\"3P68\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1190.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.90ex; vertical-align: -1.72ex; \" src=\"img/img1191.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm{senh}}x}{\\cosh x} = \\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}},$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1192.svg\" alt=\"$\\displaystyle \\quad x \\in \\mathbb{R},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1193.svg\" alt=\"$\\displaystyle {\\mathrm{ctgh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.90ex; vertical-align: -1.72ex; \" src=\"img/img1194.svg\" alt=\"$\\displaystyle = \\frac{\\cosh x}{{\\mathrm{senh}}x} = \\frac{e^{x} + e^{-x}}{e^{x} - e^{-x}},$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1195.svg\" alt=\"$\\displaystyle \\quad x \\neq 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1196.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1197.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cosh x} = \\frac{2}{e^{x} + e^{-x}},$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1192.svg\" alt=\"$\\displaystyle \\quad x \\in \\mathbb{R},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1198.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1199.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm{senh}}x} = \\frac{2}{e^{x} - e^{-x}},$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1200.svg\" alt=\"$\\displaystyle \\quad x \\neq 0.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;",
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    "title": "Capítulo 3: Igualdades exponenciais e logarítmicas",
    "section": "3.2 Método das equações diferenciais",
    "text": "3.2 Método das equações diferenciais\n\n    &lt;p class=\" unidade\" id=\"3P69\" title=\"3P69\"&gt;\n    Nesta seção, provaremos as identidades em (&lt;a href=\"#idexpo\"&gt;3.1&lt;/a&gt;) usando o método das equações diferenciais. Precisamos\n    naturalmente alguns resultados em relação às equações diferenciais. Para um estudo mais aprofundado sobre equações\n    diferenciais recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#Zill\"&gt;10&lt;/a&gt;, Zill].\n    &lt;/p&gt;\n    \n    &lt;div&gt;&lt;p style=\"text-indent: 0 !important;\"&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;&lt;/b&gt; &nbsp; \n    Uma equação diferencial ordinária, de ordem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt;, é uma equação que envolve uma variável real independente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;, uma\n    função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1201.svg\" alt=\"$y = f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; e suas derivadas &lt;!-- MATH\n     $y', y'', \\dots, y^{(n)}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1202.svg\" alt=\"$y', y'', \\dots, y^{(n)}$\" loading=\"lazy\"&gt;&lt;/span&gt;, de forma que o coeficiente de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1203.svg\" alt=\"$y^{(n)}$\" loading=\"lazy\"&gt;&lt;/span&gt; seja não nulo.\n    &lt;/p&gt;&lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P70\" title=\"3P70\"&gt;\n    São exemplos de equações diferencias ordinárias:\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P71\" title=\"3P71\"&gt;&lt;table&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1204.svg\" alt=\"$\\displaystyle y'' + y = {\\mathrm {sen}}(2x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &lt;a id=\"EDO1\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.50ex; \" src=\"img/img1205.svg\" alt=\"$\\displaystyle y \\cdot y' + x = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &lt;a id=\"EDO2\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;9&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.50ex; \" src=\"img/img1206.svg\" alt=\"$\\displaystyle y^{(4)} + xy'' +e^{x}y = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1207.svg\" alt=\"$\\displaystyle (y')^{2} + 3xy' - 4x^{2} = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &lt;a id=\"EDO4\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;10&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.53ex; vertical-align: -0.50ex; \" src=\"img/img1208.svg\" alt=\"$\\displaystyle x^{3}y''' +2x^{2}y'' - xy' + y = 12x^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &lt;a id=\"EDO5\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;11&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    \n    \n    &lt;div&gt;&lt;p style=\"text-indent: 0 !important;\"&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;&lt;/b&gt; &nbsp; \n    Uma equação diferencial ordinária linear, de ordem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt;, é uma equação diferencial que seja linear nas componentes\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1209.svg\" alt=\"$y^{(k)}$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n     $k = 0, 1, \\dots, n$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1210.svg\" alt=\"$k = 0, 1, \\dots, n$\" loading=\"lazy\"&gt;&lt;/span&gt;. É uma expressão da forma,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    a_{n}(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\cdots + a_{2}(x) y^{''} + a_{1}(x) y^{'} + a_{0}(x) y = g(x),\n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"3P72\" title=\"3P72\"&gt;\n    &lt;img style=\"height: 2.77ex; vertical-align: -0.62ex; \" src=\"img/img1211.svg\" alt=\"$\\displaystyle a_{n}(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\cdots + a_{2}(x) y^{''} + a_{1}(x) y^{'} + a_{0}(x) y = g(x), $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    sendo que as funções coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1212.svg\" alt=\"$a_{k}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n     $k = 0,1, \\dots, n$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1210.svg\" alt=\"$k = 0, 1, \\dots, n$\" loading=\"lazy\"&gt;&lt;/span&gt; e a função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img300.svg\" alt=\"$g(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, são contínuas para todo\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; em um certo intervalo de interesse &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;. Também &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1213.svg\" alt=\"$a_{n}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; é não identicamente nula em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;. Quando &lt;!-- MATH\n     $g(x) \\equiv 0$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1214.svg\" alt=\"$g(x) \\equiv 0$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n    então dizemos que a equação diferencial é homogênea.\n    &lt;/p&gt;&lt;/div&gt;\n    \n    &lt;div&gt;&lt;p style=\"text-indent: 0 !important;\"&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;&lt;/b&gt; &nbsp; \n    Qualquer função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1201.svg\" alt=\"$y = f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, definida num intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;, que satisfaz a equação diferencial neste intervalo, é dita uma\n    solução para a equação diferencial em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;&lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P73\" title=\"3P73\"&gt;\n    As funções &lt;!-- MATH\n     $y = -\\frac{1}{3}{\\mathrm {sen}}(2x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1215.svg\" alt=\"$y = -\\frac{1}{3}{\\mathrm {sen}}(2x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, &lt;!-- MATH\n     $y = -2x^{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img1216.svg\" alt=\"$y = -2x^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n     $y = x + x\\ln x + 4x^{2}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img1217.svg\" alt=\"$y = x + x\\ln x + 4x^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; (&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img637.svg\" alt=\"$x&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;), são soluções das equações\n    diferenciais (&lt;a href=\"#EDO1\"&gt;3.8&lt;/a&gt;), (&lt;a href=\"#EDO4\"&gt;3.10&lt;/a&gt;) e (&lt;a href=\"#EDO5\"&gt;3.11&lt;/a&gt;), respectivamente. A solução da equação diferencial (&lt;a href=\"#EDO2\"&gt;3.9&lt;/a&gt;)\n    é dada implicitamente por &lt;!-- MATH\n     $x^{2} + y^{2} = 4$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img1218.svg\" alt=\"$x^{2} + y^{2} = 4$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n     $-2 &lt; x &lt; 2$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1219.svg\" alt=\"$-2 &lt; x &lt; 2$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"3P74\" title=\"3P74\"&gt;\n    Embora a ideia seja bastante simples, encontrar uma solução para uma equação diferencial dada, não é tarefa simples. Os\n    métodos conhecidos nos permitem determinar soluções de uma classe muito pequena de equações diferenciais. Mesmo assim,\n    algumas destas equações não possuem solução explícita.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"3P75\" title=\"3P75\"&gt;\n    Não é do nosso interesse estudar aqui os métodos para obtenção de soluções de uma equação diferencial. Entretanto é\n    importante saber que nem sempre uma solução para uma equação diferencial é única. Podemos verificar que a função\n    &lt;!-- MATH\n     \\begin{displaymath}\n    y = k_{1} \\cos(x) + k_{2} {\\mathrm {sen}}(x),\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P76\" title=\"3P76\"&gt;\n    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1220.svg\" alt=\"$\\displaystyle y = k_{1} \\cos(x) + k_{2} {\\mathrm {sen}}(x), $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P77\" style=\"text-indent: 0 !important;\" title=\"3P77\"&gt;\n    é uma solução da equação &lt;!-- MATH\n     $y'' + y = 0$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img1221.svg\" alt=\"$y'' + y = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, para quaisquer valores reais de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img1222.svg\" alt=\"$k_{1}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1223.svg\" alt=\"$k_{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Para podermos determinar os\n    valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img1222.svg\" alt=\"$k_{1}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1223.svg\" alt=\"$k_{2}$\" loading=\"lazy\"&gt;&lt;/span&gt; são necessárias informações adicionais, chamadas de condições iniciais. Como veremos mais\n    tarde, dentro de certas hipóteses, uma equação diferencial munida de condições iniciais possui solução única.\n    &lt;/p&gt;\n    \n    &lt;div&gt;&lt;p style=\"text-indent: 0 !important;\"&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;..&lt;span class=\"arabic\"&gt;7&lt;/span&gt;&lt;/b&gt; &nbsp; \n    Um problema de valor inicial, ou PVI, consiste de uma equação diferencial ordinária, de ordem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt;, juntamente com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt;\n    restrições. Tais restrições são chamadas de condições iniciais. É um problema da forma,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    \\left\\{ \\begin{array}{l}\n    a_{n}(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\cdots + a_{2}(x) y'' + a_{1}(x) y' + a_{0}(x) y = g(x), \\\\\n    y^{(n-1)}(x_{0}) = y_{n-1}, \\quad \\dots, \\quad y''(x_{0}) = y_{2}, \\quad y'(x_{0}) = y_{1}, \\quad y(x_{0}) = y_{0}\n    \\end{array} \\right. \n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"3P78\" title=\"3P78\"&gt;\n    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1224.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l}\n    a_{n}(x) y^{(n)} + a_{n-1}(x) y^{(n-1)}...\n    ... = y_{2}, \\quad y'(x_{0}) = y_{1}, \\quad y(x_{0}) = y_{0}\n    \\end{array} \\right. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    sendo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1225.svg\" alt=\"$x_{0}$\" loading=\"lazy\"&gt;&lt;/span&gt; é um ponto de interesse no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt; e os valores &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img1226.svg\" alt=\"$y_{k}$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n     $k = 0,1, \\dots, n-1$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1227.svg\" alt=\"$k = 0,1, \\dots, n-1$\" loading=\"lazy\"&gt;&lt;/span&gt;, são\n    números reais conhecidos.\n    &lt;/p&gt;&lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P79\" title=\"3P79\"&gt;\n    Cuidado para não confundir um PVI com uma equação diferencial. Um PVI é um conjunto de uma equação diferencial\n    juntamente com condições iniciais.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"3P80\" title=\"3P80\"&gt;\n    Como exemplo, vamos agora considerar a equação diferencial de ordem 2, mencionada anteriormente &lt;!-- MATH\n     $y'' + y = 0$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img1221.svg\" alt=\"$y'' + y = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, a sua\n    “família” de soluções dada por &lt;!-- MATH\n     $y = k_{1} \\cos(x) + k_{2} {\\mathrm {sen}}(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1228.svg\" alt=\"$y = k_{1} \\cos(x) + k_{2} {\\mathrm {sen}}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; e impor duas condições iniciais que permitirão\n    determinar os valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img1222.svg\" alt=\"$k_{1}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1223.svg\" alt=\"$k_{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Tomemos o PVI,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    \\left\\{\\begin{array}{l} y'' + y = 0 \\\\y'(0) = 1, \\quad y(0) = 2, \\end{array} \\right.\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P81\" title=\"3P81\"&gt;\n    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1229.svg\" alt=\"$\\displaystyle \\left\\{\\begin{array}{l} y'' + y = 0 \\\\ y'(0) = 1, \\quad y(0) = 2, \\end{array} \\right. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P82\" style=\"text-indent: 0 !important;\" title=\"3P82\"&gt;\n    e substituindo as duas condições iniciais, temos que\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P83\" title=\"3P83\"&gt;&lt;table class=\"equation\"&gt;\n    &lt;tbody&gt;&lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1230.svg\" alt=\"$\\displaystyle 1$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1231.svg\" alt=\"$\\displaystyle = y'(0) = - k_{1} {\\mathrm {sen}}(0) + k_{2} \\cos(0) = k_{2},$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;tr&gt;\n    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1232.svg\" alt=\"$\\displaystyle 2$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1233.svg\" alt=\"$\\displaystyle = y(0) = k_{1} \\cos(0) + k_{2} {\\mathrm {sen}}(0) = k_{1},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n    &lt;p class=\" unidade\" id=\"3P84\" style=\"text-indent: 0 !important;\" title=\"3P84\"&gt;\n    o que nos leva a uma única solução do PVI dado, que é\n    &lt;!-- MATH\n     \\begin{displaymath}\n    y = 2 \\cos(x) + {\\mathrm {sen}}(x).\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P85\" title=\"3P85\"&gt;\n    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1234.svg\" alt=\"$\\displaystyle y = 2 \\cos(x) + {\\mathrm {sen}}(x). $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P86\" title=\"3P86\"&gt;\n    O próximo teorema, é a chave para o nosso objetivo. Sua demonstração é que não é do nosso interesse, pois além de não\n    ser o objetivo principal deste capítulo, é um tanto complexa e exige ferramentas que não abordamos como por exemplo o\n    teorema de ponto fixo de Banach. Desta forma, vamos omitir a sua demonstração. O leitor interessado nesta demonstração\n    poderá consultar algum texto de Equações Diferenciais (ordinárias). Sugerimos [&lt;a href=\"/trigonometria-hiperbolica/referencias#Zill\"&gt;10&lt;/a&gt;, Zill].\n    &lt;/p&gt;\n    \n    &lt;div id=\"3Teo8\" title=\"3Teo8\" class=\"unidade\"&gt;&lt;a id=\"teoPicard\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; (Teorema de Picard)  &nbsp; \n    &lt;i&gt;Se as funções &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1212.svg\" alt=\"$a_{k}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n     $k = 0,1,\\dots, n$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1210.svg\" alt=\"$k = 0, 1, \\dots, n$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img300.svg\" alt=\"$g(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, forem contínuas em um intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;, com &lt;!-- MATH\n     $a_{n}(x)\n    \\neq 0$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1235.svg\" alt=\"$a_{n}(x)\n    \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1133.svg\" alt=\"$x \\in I$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1225.svg\" alt=\"$x_{0}$\" loading=\"lazy\"&gt;&lt;/span&gt; é um ponto de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img529.svg\" alt=\"$I$\" loading=\"lazy\"&gt;&lt;/span&gt;, então o PVI\n    &lt;/i&gt;&lt;!-- MATH\n     \\begin{displaymath}\n    \\left\\{ \\begin{array}{l}\n    a_{n}(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + \\cdots + a_{2}(x) y'' + a_{1}(x) y' + a_{0}(x) y = g(x), \\\\\n    y^{(n-1)}(x_{0}) = y_{n-1}, \\quad \\dots, \\quad y''(x_{0})=y_{2}, \\quad y'(x_{0}) = y_{1}, \\quad y(x_{0}) = y_{0}\n    \\end{array} \\right. \n    \\end{displaymath}\n     --&gt;\n    \n    &lt;div class=\"mathdisplay unidade\" id=\"3P87\" title=\"3P87\"&gt;\n    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1224.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l}\n    a_{n}(x) y^{(n)} + a_{n-1}(x) y^{(n-1)}...\n    ... = y_{2}, \\quad y'(x_{0}) = y_{1}, \\quad y(x_{0}) = y_{0}\n    \\end{array} \\right. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;i&gt;\n    possui uma única solução &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1201.svg\" alt=\"$y = f(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, neste intervalo.\n    &lt;/i&gt;&lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P88\" title=\"3P88\"&gt;\n    Agora estamos prontos para estabelecer as identidades mencionadas no início deste capítulo. Para isto, consideremos\n    primeiro o problema de valor inicial, definido em &lt;!-- MATH\n     $I = \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1236.svg\" alt=\"$I = \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    \\left\\{ \\begin{array}{l}\n    y'' - y  = 0 \\\\\n    y'(0) = 0, \\quad y(0) = 1.\n    \\end{array} \\right. \n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P89\" title=\"3P89\"&gt;\n    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1237.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l}\n    y'' - y = 0 \\\\\n    y'(0) = 0, \\quad y(0) = 1.\n    \\end{array} \\right. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P90\" title=\"3P90\"&gt;\n    Notemos que a função\n    &lt;!-- MATH\n     \\begin{displaymath}\n    y_{1} = \\frac{1}{2} e^{x} + \\frac{1}{2} e^{-x} = \\frac{e^{x} + e^{-x}}{2}\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P91\" title=\"3P91\"&gt;\n    &lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1238.svg\" alt=\"$\\displaystyle y_{1} = \\frac{1}{2} e^{x} + \\frac{1}{2} e^{-x} = \\frac{e^{x} + e^{-x}}{2} $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P92\" style=\"text-indent: 0 !important;\" title=\"3P92\"&gt;\n    é solução do PVI dado. Mas, do que vimos nos capítulos anteriores a respeito das funções trigonométricas hiperbólicas,\n    a função &lt;!-- MATH\n     $y_{2} = \\cosh(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1239.svg\" alt=\"$y_{2} = \\cosh(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;, satisfaz a equação diferencial, pois\n    &lt;!-- MATH\n     \\begin{displaymath}\n    (y_{2})'' - y_{2} = (\\cosh(x))'' - \\cosh(x) = \\cosh(x) - \\cosh(x) = 0,\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P93\" title=\"3P93\"&gt;\n    &lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1240.svg\" alt=\"$\\displaystyle (y_{2})'' - y_{2} = (\\cosh(x))'' - \\cosh(x) = \\cosh(x) - \\cosh(x) = 0, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P94\" style=\"text-indent: 0 !important;\" title=\"3P94\"&gt;\n    para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; real e, além disso, &lt;!-- MATH\n     $y_{2} = \\cosh(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1239.svg\" alt=\"$y_{2} = \\cosh(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; satisfaz as duas condições iniciais,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    y_{2}(0) = \\cosh(0) = 1, \\qquad \\text{e} \\qquad y_{2}'(0) = {\\mathrm{senh}}(0) = 0,\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P95\" title=\"3P95\"&gt;\n    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1241.svg\" alt=\"$\\displaystyle y_{2}(0) = \\cosh(0) = 1,$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.45ex; vertical-align: -0.63ex; \" src=\"img/img1242.svg\" alt=\"$\\displaystyle \\qquad y_{2}'(0) = {\\mathrm{senh}}(0) = 0, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P96\" style=\"text-indent: 0 !important;\" title=\"3P96\"&gt;\n    donde temos que &lt;!-- MATH\n     $y_{2} = \\cosh(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1239.svg\" alt=\"$y_{2} = \\cosh(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; é também uma solução do PVI. Mas o Teorema de Picard, garante que a solução deste\n    PVI é única e, portanto, as duas soluções coincidem para todo &lt;!-- MATH\n     $x \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    \\cosh(x) = y_{2} = y_{1} = \\frac{e^{x} + e^{-x}}{2},\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P97\" title=\"3P97\"&gt;\n    &lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1243.svg\" alt=\"$\\displaystyle \\cosh(x) = y_{2} = y_{1} = \\frac{e^{x} + e^{-x}}{2}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P98\" style=\"text-indent: 0 !important;\" title=\"3P98\"&gt;\n    qualquer que seja &lt;!-- MATH\n     $x \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;p class=\" unidade\" id=\"3P99\" title=\"3P99\"&gt;\n    Para a segunda fórmula em (&lt;a href=\"#idexpo\"&gt;3.1&lt;/a&gt;), consideremos outro problema de valor inicial, também definido em &lt;!-- MATH\n     $I = \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1236.svg\" alt=\"$I = \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    \\left\\{ \\begin{array}{l}\n    y'' - y  = 0 \\\\\n    y'(0) = 1, \\quad y(0) = 0.\n    \\end{array} \\right. \n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P100\" title=\"3P100\"&gt;\n    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1244.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l}\n    y'' - y = 0 \\\\\n    y'(0) = 1, \\quad y(0) = 0.\n    \\end{array} \\right. $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;\n    \n    &lt;p class=\" unidade\" id=\"3P101\" title=\"3P101\"&gt;\n    Observe que, comparando com o PVI anterior, apenas trocamos as condições iniciais. Nestes termos, a função\n    &lt;!-- MATH\n     \\begin{displaymath}\n    y_{1} = \\frac{1}{2} e^{x} - \\frac{1}{2} e^{-x} = \\frac{e^{x} - e^{-x}}{2},\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P102\" title=\"3P102\"&gt;\n    &lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1245.svg\" alt=\"$\\displaystyle y_{1} = \\frac{1}{2} e^{x} - \\frac{1}{2} e^{-x} = \\frac{e^{x} - e^{-x}}{2}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P103\" style=\"text-indent: 0 !important;\" title=\"3P103\"&gt;\n    é solução deste novo PVI. Entretanto, do que vimos nos capítulos anteriores, a função &lt;!-- MATH\n     $y_{2} = {\\mathrm{senh}}(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1246.svg\" alt=\"$y_{2} = {\\mathrm{senh}}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; também\n    satisfaz a equação diferencial,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    (y_{2})'' - y_{2} = ({\\mathrm{senh}}(x))'' - {\\mathrm{senh}}(x) = {\\mathrm{senh}}(x) - {\\mathrm{senh}}(x) = 0,\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P104\" title=\"3P104\"&gt;\n    &lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1247.svg\" alt=\"$\\displaystyle (y_{2})'' - y_{2} = ({\\mathrm{senh}}(x))'' - {\\mathrm{senh}}(x) = {\\mathrm{senh}}(x) - {\\mathrm{senh}}(x) = 0, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P105\" style=\"text-indent: 0 !important;\" title=\"3P105\"&gt;\n    para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; real e as duas condições inicias,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    y_{2}(0) = {\\mathrm{senh}}(0) = 0, \\qquad \\text{e} \\qquad y_{2}'(0) = \\cosh(0) = 1,\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P106\" title=\"3P106\"&gt;\n    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1248.svg\" alt=\"$\\displaystyle y_{2}(0) = {\\mathrm{senh}}(0) = 0,$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.45ex; vertical-align: -0.63ex; \" src=\"img/img1249.svg\" alt=\"$\\displaystyle \\qquad y_{2}'(0) = \\cosh(0) = 1, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P107\" style=\"text-indent: 0 !important;\" title=\"3P107\"&gt;\n    e, dessa forma, &lt;!-- MATH\n     $y_{2} = {\\mathrm{senh}}(x)$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1246.svg\" alt=\"$y_{2} = {\\mathrm{senh}}(x)$\" loading=\"lazy\"&gt;&lt;/span&gt; também é solução do PVI, para todo &lt;!-- MATH\n     $x \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Do Teorema de Picard, segue que as\n    duas soluções coincidem para todo &lt;!-- MATH\n     $x \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, isto é,\n    &lt;!-- MATH\n     \\begin{displaymath}\n    {\\mathrm{senh}}(x) = y_{2} = y_{1} = \\frac{e^{x} - e^{-x}}{2},\n    \\end{displaymath}\n     --&gt;\n    &lt;/p&gt;\n    &lt;div class=\"mathdisplay unidade\" id=\"3P108\" title=\"3P108\"&gt;\n    &lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1250.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}(x) = y_{2} = y_{1} = \\frac{e^{x} - e^{-x}}{2}, $\" loading=\"lazy\"&gt;\n    &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P109\" style=\"text-indent: 0 !important;\" title=\"3P109\"&gt;\n    para todo &lt;!-- MATH\n     $x \\in \\mathbb{R}$\n     --&gt;\n    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n    &lt;/p&gt;\n    &lt;a id=\"secforlog\"&gt;&lt;/a&gt;",
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      "Capítulo 3: Igualdades exponenciais e logarítmicas"
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    "title": "Capítulo 3: Igualdades exponenciais e logarítmicas",
    "section": "3.3 As fórmulas logarítmicas",
    "text": "3.3 As fórmulas logarítmicas\n\n        &lt;p class=\" unidade\" id=\"3P110\" title=\"3P110\"&gt;\n        Esta seção é dedicada à obtenção das igualdades logarítmicas em (&lt;a href=\"#idlog\"&gt;3.2&lt;/a&gt;), além das igualdades correspondentes às\n        outras quatro funções trigonométricas hiperbólicas inversas.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P111\" title=\"3P111\"&gt;\n        Dados &lt;!-- MATH\n         $x, y \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.04ex; vertical-align: -0.50ex; \" src=\"img/img1251.svg\" alt=\"$x, y \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; de forma que &lt;!-- MATH\n         $y = {\\mathrm{senh}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1252.svg\" alt=\"$y = {\\mathrm{senh}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;, já sabemos que é válida a relação\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = {\\mathrm{senh}}y = \\frac{e^{y} - e^{-y}}{2}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P112\" title=\"3P112\"&gt;\n        &lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1253.svg\" alt=\"$\\displaystyle x = {\\mathrm{senh}}y = \\frac{e^{y} - e^{-y}}{2}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P113\" title=\"3P113\"&gt;\n        Vamos isolar &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; no segundo membro e obter uma expressão para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; em termos da variável independente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt;. A igualdade\n        anterior, nos leva a\n        &lt;!-- MATH\n         \\begin{displaymath}\n        2x = e^{y} - e^{-y}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P114\" title=\"3P114\"&gt;\n        &lt;img style=\"height: 2.13ex; vertical-align: -0.27ex; \" src=\"img/img1254.svg\" alt=\"$\\displaystyle 2x = e^{y} - e^{-y}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P115\" title=\"3P115\"&gt;\n        Multiplicando ambos os membros por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; e reorganizando os termos temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (e^{y})^{2} - 2xe^{y} - 1 = 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P116\" title=\"3P116\"&gt;\n        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1256.svg\" alt=\"$\\displaystyle (e^{y})^{2} - 2xe^{y} - 1 = 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P117\" style=\"text-indent: 0 !important;\" title=\"3P117\"&gt;\n        que é uma equação quadrática na expressão &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt;. As soluções desta equação quadrática, são dadas por\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{2x \\pm \\sqrt{4x^{2} + 4} }{2} = x \\pm \\sqrt{x^{2} + 1}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P118\" title=\"3P118\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1257.svg\" alt=\"$\\displaystyle e^{y} = \\frac{2x \\pm \\sqrt{4x^{2} + 4} }{2} = x \\pm \\sqrt{x^{2} + 1}.$\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P119\" title=\"3P119\"&gt;\n        Temos que descartar uma das soluções porque o lado esquerdo da igualdade acima é sempre positivo e o termo &lt;!-- MATH\n         $x -\n        \\sqrt{x^{2}+1}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1258.svg\" alt=\"$x -\n        \\sqrt{x^{2}+1}$\" loading=\"lazy\"&gt;&lt;/span&gt; é sempre negativo já que &lt;!-- MATH\n         $\\sqrt{x^{2}+1} &gt; x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1259.svg\" alt=\"$\\sqrt{x^{2}+1} &gt; x$\" loading=\"lazy\"&gt;&lt;/span&gt;. Tomando então a solução positiva temos &lt;!-- MATH\n         $e^{y} = x +\n        \\sqrt{x^{2} + 1}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1260.svg\" alt=\"$e^{y} = x +\n        \\sqrt{x^{2} + 1}$\" loading=\"lazy\"&gt;&lt;/span&gt; e aplicando logaritmo (natural) em ambos os membros,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm{senh}}^{-1} x = y = \\ln(x + \\sqrt{x^{2} + 1}),\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P120\" title=\"3P120\"&gt;\n        &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img1261.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}^{-1} x = y = \\ln(x + \\sqrt{x^{2} + 1}), $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P121\" style=\"text-indent: 0 !important;\" title=\"3P121\"&gt;\n        para todo &lt;!-- MATH\n         $x \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P122\" title=\"3P122\"&gt;\n        Para o cosseno hiperbólico inverso, consideramos &lt;!-- MATH\n         $y = \\cosh^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1262.svg\" alt=\"$y = \\cosh^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt; e a relação inversa &lt;!-- MATH\n         $x = \\cosh y$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1263.svg\" alt=\"$x = \\cosh y$\" loading=\"lazy\"&gt;&lt;/span&gt;, válida para\n        todos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1264.svg\" alt=\"$x \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1265.svg\" alt=\"$y \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como antes, tomemos a identidade\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = \\cosh y = \\frac{e^{y} + e^{-y}}{2},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P123\" title=\"3P123\"&gt;\n        &lt;img style=\"height: 4.72ex; vertical-align: -1.55ex; \" src=\"img/img1266.svg\" alt=\"$\\displaystyle x = \\cosh y = \\frac{e^{y} + e^{-y}}{2}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P124\" style=\"text-indent: 0 !important;\" title=\"3P124\"&gt;\n        e vamos isolar &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\" loading=\"lazy\"&gt;&lt;/span&gt; no segundo membro. De forma análoga ao caso anterior, multiplicamos os dois membros por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1267.svg\" alt=\"$2e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        reorganizamos os termos e chegamos a\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (e^{y})^{2} - 2xe^{y} + 1 = 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P125\" title=\"3P125\"&gt;\n        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1268.svg\" alt=\"$\\displaystyle (e^{y})^{2} - 2xe^{y} + 1 = 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P126\" style=\"text-indent: 0 !important;\" title=\"3P126\"&gt;\n        e resolvendo esta equação quadrática em termos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{2x \\pm \\sqrt{4x^{2} - 4}}{2} = x \\pm \\sqrt{x^{2}-1}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P127\" title=\"3P127\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1269.svg\" alt=\"$\\displaystyle e^{y} = \\frac{2x \\pm \\sqrt{4x^{2} - 4}}{2} = x \\pm \\sqrt{x^{2}-1}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P128\" title=\"3P128\"&gt;\n        Observamos agora que os dois termos a que se refere o segundo membro são positivos e, portanto, não há impossibilidades\n        matemáticas para aplicar o logaritmo. Entretanto lembremos que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1265.svg\" alt=\"$y \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e isto siginifica que &lt;!-- MATH\n         $e^{y} \\geq 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.95ex; vertical-align: -0.38ex; \" src=\"img/img1270.svg\" alt=\"$e^{y} \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mas\n        para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1264.svg\" alt=\"$x \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt; temos que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = 1+\\sqrt{(x-1)^{2}} = 1+\\sqrt{x^{2}-2x+1} \\leq 1+\\sqrt{x^{2}-2+1} = 1+\\sqrt{x^{2}-1},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P129\" title=\"3P129\"&gt;\n        &lt;img style=\"height: 4.07ex; vertical-align: -1.22ex; \" src=\"img/img1271.svg\" alt=\"$\\displaystyle x = 1+\\sqrt{(x-1)^{2}} = 1+\\sqrt{x^{2}-2x+1} \\leq 1+\\sqrt{x^{2}-2+1} = 1+\\sqrt{x^{2}-1}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P130\" style=\"text-indent: 0 !important;\" title=\"3P130\"&gt;\n        e desta forma &lt;!-- MATH\n         $x - \\sqrt{x^{2}-1} \\leq 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.46ex; vertical-align: -0.38ex; \" src=\"img/img1272.svg\" alt=\"$x - \\sqrt{x^{2}-1} \\leq 1$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P131\" title=\"3P131\"&gt;\n        Descartando esta inconsistência, tomamos &lt;!-- MATH\n         $e^{y} = x + \\sqrt{x^{2}-1}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1273.svg\" alt=\"$e^{y} = x + \\sqrt{x^{2}-1}$\" loading=\"lazy\"&gt;&lt;/span&gt; e aplicando o logaritmo em ambos os membros, temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\cosh^{-1} x = y = \\ln(x + \\sqrt{x^{2}-1}).\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P132\" title=\"3P132\"&gt;\n        &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img1274.svg\" alt=\"$\\displaystyle \\cosh^{-1} x = y = \\ln(x + \\sqrt{x^{2}-1}). $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P133\" title=\"3P133\"&gt;\n        Já provamos as duas fórmulas indicadas no início deste capítulo. Contudo, vamos completar o trabalho e obter as\n        fórmulas em termos do logaritmo para as demais funções trigonométricas hiperbólicas.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P134\" title=\"3P134\"&gt;\n        Consideremos &lt;!-- MATH\n         $y = {\\mathrm {tgh}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1275.svg\" alt=\"$y = {\\mathrm {tgh}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt; e a relação inversa\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = {\\mathrm {tgh}}y = \\frac{e^{y} - e^{-y}}{e^{y} + e^{-y}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P135\" title=\"3P135\"&gt;\n        &lt;img style=\"height: 4.90ex; vertical-align: -1.72ex; \" src=\"img/img1276.svg\" alt=\"$\\displaystyle x = {\\mathrm {tgh}}y = \\frac{e^{y} - e^{-y}}{e^{y} + e^{-y}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P136\" style=\"text-indent: 0 !important;\" title=\"3P136\"&gt;\n        válida para &lt;!-- MATH\n         $x \\in (-1,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1277.svg\" alt=\"$x \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $y \\in \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.04ex; vertical-align: -0.50ex; \" src=\"img/img1278.svg\" alt=\"$y \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Organizando os termos temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        xe^{y} + xe^{-y} = e^{y} - e^{-y},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P137\" title=\"3P137\"&gt;\n        &lt;img style=\"height: 2.13ex; vertical-align: -0.27ex; \" src=\"img/img1279.svg\" alt=\"$\\displaystyle xe^{y} + xe^{-y} = e^{y} - e^{-y}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P138\" style=\"text-indent: 0 !important;\" title=\"3P138\"&gt;\n        e multiplicando ambos os membros por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; e reorganizando em forma de equação quadrática, chegamos a\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (1-x)(e^{y})^{2} - (1+x) = 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P139\" title=\"3P139\"&gt;\n        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1280.svg\" alt=\"$\\displaystyle (1-x)(e^{y})^{2} - (1+x) = 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P140\" style=\"text-indent: 0 !important;\" title=\"3P140\"&gt;\n        que resolvida em termos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; fornece\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\pm \\sqrt{ \\tfrac{1+x}{1-x} } = \\pm \\left( \\tfrac{1+x}{1-x} \\right)^{\\frac{1}{2}}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P141\" title=\"3P141\"&gt;\n        &lt;img style=\"height: 4.08ex; vertical-align: -1.28ex; \" src=\"img/img1281.svg\" alt=\"$\\displaystyle e^{y} = \\pm \\sqrt{ \\tfrac{1+x}{1-x} } = \\pm \\left( \\tfrac{1+x}{1-x} \\right)^{\\frac{1}{2}}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P142\" title=\"3P142\"&gt;\n        Como o primeiro membro é sempre positivo, então descartamos a solução negativa. Observemos também que como &lt;!-- MATH\n         $x \\in\n        (-1,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1277.svg\" alt=\"$x \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt; então a fração dentro da raiz quadrada é sempre positiva, o que não acarreta mais inconsistências. Aplicando\n        então o logaritmo, temos que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm {tgh}}^{-1} x = y = \\ln \\left( \\tfrac{1+x}{1-x} \\right)^{\\frac{1}{2}} = \\frac{1}{2} \\ln \\left( \\tfrac{1+x}{1-x} \\right),\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P143\" title=\"3P143\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1282.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}^{-1} x = y = \\ln \\left( \\tfrac{1+x}{1-x} \\right)^{\\frac{1}{2}} = \\frac{1}{2} \\ln \\left( \\tfrac{1+x}{1-x} \\right), $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P144\" style=\"text-indent: 0 !important;\" title=\"3P144\"&gt;\n        para todo &lt;!-- MATH\n         $x \\in (-1,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1277.svg\" alt=\"$x \\in (-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P145\" title=\"3P145\"&gt;\n        Agora a cotangente hiperbólica inversa. Tomamos &lt;!-- MATH\n         $y = {\\mathrm{ctgh}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1283.svg\" alt=\"$y = {\\mathrm{ctgh}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n         $x \\in (-\\infty,-1) \\cup (1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1284.svg\" alt=\"$x \\in (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;, com\n        &lt;!-- MATH\n         $y \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1285.svg\" alt=\"$y \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; e então\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = {\\mathrm{ctgh}}y = \\frac{e^{y} + e^{-y}}{e^{y} - e^{-y}}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P146\" title=\"3P146\"&gt;\n        &lt;img style=\"height: 4.90ex; vertical-align: -1.72ex; \" src=\"img/img1286.svg\" alt=\"$\\displaystyle x = {\\mathrm{ctgh}}y = \\frac{e^{y} + e^{-y}}{e^{y} - e^{-y}}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P147\" title=\"3P147\"&gt;\n        Como no caso da tangente hiperbólica, reorganizamos os termos e multiplicamos por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; ambos os membros e chegamos a\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (x-1)(e^{y})^{2} - (x+1) = 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P148\" title=\"3P148\"&gt;\n        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1287.svg\" alt=\"$\\displaystyle (x-1)(e^{y})^{2} - (x+1) = 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P149\" style=\"text-indent: 0 !important;\" title=\"3P149\"&gt;\n        e resolvendo esta equação quadrática em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\pm \\sqrt{ \\tfrac{x+1}{x-1} } = \\pm \\left( \\tfrac{x+1}{x-1} \\right)^{\\frac{1}{2}}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P150\" title=\"3P150\"&gt;\n        &lt;img style=\"height: 4.08ex; vertical-align: -1.28ex; \" src=\"img/img1288.svg\" alt=\"$\\displaystyle e^{y} = \\pm \\sqrt{ \\tfrac{x+1}{x-1} } = \\pm \\left( \\tfrac{x+1}{x-1} \\right)^{\\frac{1}{2}}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P151\" title=\"3P151\"&gt;\n        Note que a fração dentro da raiz quadrada é sempre positiva para &lt;!-- MATH\n         $x \\in (-\\infty,-1) \\cup (1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1284.svg\" alt=\"$x \\in (-\\infty,-1) \\cup (1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. De fato, o\n        numerador e o denominador são ambos negativos no intervalo &lt;!-- MATH\n         $(-\\infty,-1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1289.svg\" alt=\"$(-\\infty,-1)$\" loading=\"lazy\"&gt;&lt;/span&gt; e são ambos positivos no intervalo\n        &lt;!-- MATH\n         $(1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1290.svg\" alt=\"$(1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Vamos descartar a solução negativa, pois o lado esquerdo da igualdade é sempre positivo. Assim, tomando a\n        solução positiva e aplicando logaritmo em ambos os membros, vem\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm{ctgh}}^{-1} x = y = \\ln \\left( \\tfrac{x+1}{x-1} \\right)^{\\frac{1}{2}} = \\frac{1}{2} \\ln \\left( \\tfrac{x+1}{x-1} \\right).\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P152\" title=\"3P152\"&gt;\n        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1291.svg\" alt=\"$\\displaystyle {\\mathrm{ctgh}}^{-1} x = y = \\ln \\left( \\tfrac{x+1}{x-1} \\right)^{\\frac{1}{2}} = \\frac{1}{2} \\ln \\left( \\tfrac{x+1}{x-1} \\right). $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P153\" title=\"3P153\"&gt;\n        Para a secante hiperbólica inversa, fazendo &lt;!-- MATH\n         $y = {\\mathrm{sech}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1292.svg\" alt=\"$y = {\\mathrm{sech}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n         $x \\in (0,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1293.svg\" alt=\"$x \\in (0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;, com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1265.svg\" alt=\"$y \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = {\\mathrm{sech}}y = \\frac{2}{e^{y} + e^{-y}}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P154\" title=\"3P154\"&gt;\n        &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1294.svg\" alt=\"$\\displaystyle x = {\\mathrm{sech}}y = \\frac{2}{e^{y} + e^{-y}}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P155\" title=\"3P155\"&gt;\n        Após reorganização dos termos e multiplicação por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos a equação quadrática\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x(e^{y})^{2} - 2e^{y} + x = 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P156\" title=\"3P156\"&gt;\n        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1295.svg\" alt=\"$\\displaystyle x(e^{y})^{2} - 2e^{y} + x = 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P157\" style=\"text-indent: 0 !important;\" title=\"3P157\"&gt;\n        que resolvida em termos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt; nos traz\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{2 \\pm \\sqrt{4 - 4x^{2}} }{2x} = \\frac{1 \\pm \\sqrt{1-x^{2}} }{x}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P158\" title=\"3P158\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1296.svg\" alt=\"$\\displaystyle e^{y} = \\frac{2 \\pm \\sqrt{4 - 4x^{2}} }{2x} = \\frac{1 \\pm \\sqrt{1-x^{2}} }{x}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P159\" title=\"3P159\"&gt;\n        Notemos que para &lt;!-- MATH\n         $x \\in (0,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1293.svg\" alt=\"$x \\in (0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; ocorre &lt;!-- MATH\n         $1-x^{2} \\geq 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.38ex; \" src=\"img/img1297.svg\" alt=\"$1-x^{2} \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, não temos problemas com a raiz quadrada. Entretanto,\n        para &lt;!-- MATH\n         $x \\in (0,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1293.svg\" alt=\"$x \\in (0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; temos &lt;!-- MATH\n         $(1-x) \\geq 0$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1298.svg\" alt=\"$(1-x) \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e então\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1-x = \\sqrt{(1-x)^{2}} = \\sqrt{(1-x)(1-x)} \\leq \\sqrt{(1-x)(1+x)} = \\sqrt{1-x^{2}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P160\" title=\"3P160\"&gt;\n        &lt;img style=\"height: 4.07ex; vertical-align: -1.22ex; \" src=\"img/img1299.svg\" alt=\"$\\displaystyle 1-x = \\sqrt{(1-x)^{2}} = \\sqrt{(1-x)(1-x)} \\leq \\sqrt{(1-x)(1+x)} = \\sqrt{1-x^{2}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P161\" style=\"text-indent: 0 !important;\" title=\"3P161\"&gt;\n        donde temos que &lt;!-- MATH\n         $1 - \\sqrt{1-x^{2}} \\leq x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.46ex; vertical-align: -0.38ex; \" src=\"img/img1300.svg\" alt=\"$1 - \\sqrt{1-x^{2}} \\leq x$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, &lt;!-- MATH\n         $\\frac{1 - \\sqrt{1-x^{2}} }{x} \\leq 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.22ex; vertical-align: -0.83ex; \" src=\"img/img1301.svg\" alt=\"$\\frac{1 - \\sqrt{1-x^{2}} }{x} \\leq 1$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mas isto é inconsistente\n        com o primeiro membro &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt;, que é maior ou igual a 1, já que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1265.svg\" alt=\"$y \\geq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Só não seria inconsistente caso os dois\n        termos fossem iguais a 1, isto é &lt;!-- MATH\n         $e^{y} = 1 = \\frac{1 - \\sqrt{1-x^{2}} }{x}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.22ex; vertical-align: -0.83ex; \" src=\"img/img1302.svg\" alt=\"$e^{y} = 1 = \\frac{1 - \\sqrt{1-x^{2}} }{x}$\" loading=\"lazy\"&gt;&lt;/span&gt;, que somente ocorre se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1303.svg\" alt=\"$y = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        Mas a igualdade &lt;!-- MATH\n         $e^{y} = \\frac{1 + \\sqrt{1-x^{2}} }{x}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.22ex; vertical-align: -0.83ex; \" src=\"img/img1305.svg\" alt=\"$e^{y} = \\frac{1 + \\sqrt{1-x^{2}} }{x}$\" loading=\"lazy\"&gt;&lt;/span&gt; também se verifica para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1303.svg\" alt=\"$y = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, podemos\n        descartar totalmente a solução &lt;!-- MATH\n         $e^{y} = \\frac{1-\\sqrt{1-x^{2}} }{x}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.22ex; vertical-align: -0.83ex; \" src=\"img/img1306.svg\" alt=\"$e^{y} = \\frac{1-\\sqrt{1-x^{2}} }{x}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P162\" title=\"3P162\"&gt;\n        Tomando então a solução que não apresenta inconsistências, tomamos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{1 + \\sqrt{1-x^{2}} }{x},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P163\" title=\"3P163\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1307.svg\" alt=\"$\\displaystyle e^{y} = \\frac{1 + \\sqrt{1-x^{2}} }{x}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P164\" style=\"text-indent: 0 !important;\" title=\"3P164\"&gt;\n        e aplicando o logaritmo, temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm{sech}}^{-1} x = y = \\ln \\left( \\frac{1}{x} + \\frac{\\sqrt{1-x^{2}}}{x} \\right).\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P165\" title=\"3P165\"&gt;\n        &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1308.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}^{-1} x = y = \\ln \\left( \\frac{1}{x} + \\frac{\\sqrt{1-x^{2}}}{x} \\right). $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P166\" title=\"3P166\"&gt;\n        Finalmente, considerando &lt;!-- MATH\n         $y = {\\mathrm{csch}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.51ex; vertical-align: -0.50ex; \" src=\"img/img1309.svg\" alt=\"$y = {\\mathrm{csch}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt; e a relação inversa\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x = {\\mathrm{csch}}y = \\frac{2}{e^{y} - e^{-y}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P167\" title=\"3P167\"&gt;\n        &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1310.svg\" alt=\"$\\displaystyle x = {\\mathrm{csch}}y = \\frac{2}{e^{y} - e^{-y}}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P168\" style=\"text-indent: 0 !important;\" title=\"3P168\"&gt;\n        válida para todos &lt;!-- MATH\n         $x \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1311.svg\" alt=\"$x \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $y \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1285.svg\" alt=\"$y \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Procedendo como no caso da secante, obtemos a equação quadrática\n        &lt;!-- MATH\n         \\begin{displaymath}\n        x(e^{y})^{2} - 2e^{y} - x = 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P169\" title=\"3P169\"&gt;\n        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1312.svg\" alt=\"$\\displaystyle x(e^{y})^{2} - 2e^{y} - x = 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P170\" style=\"text-indent: 0 !important;\" title=\"3P170\"&gt;\n        que resolvida em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1255.svg\" alt=\"$e^{y}$\" loading=\"lazy\"&gt;&lt;/span&gt;, nos fornece\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{2 \\pm \\sqrt{4 + 4x^{2}} }{2x} = \\frac{1 \\pm \\sqrt{1 + x^{2}} }{x}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P171\" title=\"3P171\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1313.svg\" alt=\"$\\displaystyle e^{y} = \\frac{2 \\pm \\sqrt{4 + 4x^{2}} }{2x} = \\frac{1 \\pm \\sqrt{1 + x^{2}} }{x}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P172\" title=\"3P172\"&gt;\n        Notemos que como antes, queremos que o membro da direita seja positivo, pois o da esquerda o é. O termo\n        &lt;!-- MATH\n         $\\sqrt{1+x^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1314.svg\" alt=\"$\\sqrt{1+x^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt; é sempre maior que 1. O numerador assume portanto valores positivos considerando &lt;!-- MATH\n         $1 + \\sqrt{1+x^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1315.svg\" alt=\"$1 + \\sqrt{1+x^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n        e valores negativos considerando &lt;!-- MATH\n         $1 - \\sqrt{1+x^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.31ex; \" src=\"img/img1316.svg\" alt=\"$1 - \\sqrt{1+x^{2}}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Mas como &lt;!-- MATH\n         $x \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1311.svg\" alt=\"$x \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt; temos que o denominador também assume\n        valores positivos e valores negativos. Então se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1317.svg\" alt=\"$x &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, devemos considerar a solução\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{1 - \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} - \\frac{\\sqrt{1+x^{2}}}{x} &gt; 0,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P173\" title=\"3P173\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1318.svg\" alt=\"$\\displaystyle e^{y} = \\frac{1 - \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} - \\frac{\\sqrt{1+x^{2}}}{x} &gt; 0, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P174\" style=\"text-indent: 0 !important;\" title=\"3P174\"&gt;\n        e se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img637.svg\" alt=\"$x&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt;, devemos considerar a solução\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{1 + \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} + \\frac{\\sqrt{1+x^{2}}}{x} &gt; 0.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P175\" title=\"3P175\"&gt;\n        &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1319.svg\" alt=\"$\\displaystyle e^{y} = \\frac{1 + \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} + \\frac{\\sqrt{1+x^{2}}}{x} &gt; 0. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P176\" title=\"3P176\"&gt;\n        Podemos ainda obter uma única expressão válida para os dois casos. Observe que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img637.svg\" alt=\"$x&gt;0$\" loading=\"lazy\"&gt;&lt;/span&gt; podemos escrever\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{1}{x} + \\frac{ \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} + \\frac{ \\sqrt{1 + x^{2}} }{|x|},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P177\" title=\"3P177\"&gt;\n        &lt;img style=\"height: 5.57ex; vertical-align: -2.07ex; \" src=\"img/img1320.svg\" alt=\"$\\displaystyle e^{y} = \\frac{1}{x} + \\frac{ \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} + \\frac{ \\sqrt{1 + x^{2}} }{\\vert x\\vert}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P178\" style=\"text-indent: 0 !important;\" title=\"3P178\"&gt;\n        e se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1317.svg\" alt=\"$x &lt; 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, podemos escrever\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{1}{x} - \\frac{ \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} + \\frac{\\sqrt{1 + x^{2}}}{-x} = \\frac{1}{x} + \\frac{\\sqrt{1 + x^{2}}}{|x|}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P179\" title=\"3P179\"&gt;\n        &lt;img style=\"height: 5.57ex; vertical-align: -2.07ex; \" src=\"img/img1321.svg\" alt=\"$\\displaystyle e^{y} = \\frac{1}{x} - \\frac{ \\sqrt{1 + x^{2}} }{x} = \\frac{1}{x} ...\n        ...c{\\sqrt{1 + x^{2}}}{-x} = \\frac{1}{x} + \\frac{\\sqrt{1 + x^{2}}}{\\vert x\\vert}. $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P180\" title=\"3P180\"&gt;\n        Assim, para qualquer &lt;!-- MATH\n         $x \\in \\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1311.svg\" alt=\"$x \\in \\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;, escrevemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        e^{y} = \\frac{1}{x} + \\frac{\\sqrt{1 + x^{2}} }{|x|},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P181\" title=\"3P181\"&gt;\n        &lt;img style=\"height: 5.57ex; vertical-align: -2.07ex; \" src=\"img/img1322.svg\" alt=\"$\\displaystyle e^{y} = \\frac{1}{x} + \\frac{\\sqrt{1 + x^{2}} }{\\vert x\\vert}, $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P182\" style=\"text-indent: 0 !important;\" title=\"3P182\"&gt;\n        e aplicando o logaritmo, temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm{csch}}^{-1} x = y = \\ln \\left( \\tfrac{1}{x} + \\tfrac{\\sqrt{1 + x^{2}} }{|x|} \\right),\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"3P183\" title=\"3P183\"&gt;\n        &lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1323.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}^{-1} x = y = \\ln \\left( \\tfrac{1}{x} + \\tfrac{\\sqrt{1 + x^{2}} }{\\vert x\\vert} \\right), $\" loading=\"lazy\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P184\" style=\"text-indent: 0 !important;\" title=\"3P184\"&gt;\n        que é válida para todo &lt;!-- MATH\n         $x \\in \\mathbb{R}^{*}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.75ex; vertical-align: -0.18ex; \" src=\"img/img1324.svg\" alt=\"$x \\in \\mathbb{R}^{*}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"3P185\" title=\"3P185\"&gt;\n        A tabela abaixo reúne as fórmulas desta seção.\n        \n        &lt;/p&gt;\n        &lt;div class=\"CENTER\"&gt;&lt;a id=\"3995\"&gt;&lt;/a&gt;\n        &lt;table&gt;\n        &lt;caption&gt;&lt;strong&gt;Tabela 3.1:&lt;/strong&gt;\n        Fórmulas logarítmicas para as funções trigonométricas hiperbólicas inversas.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n        &lt;div class=\"CENTER\"&gt;\n        &lt;table class=\"PAD  BORDER\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"LEFT\"&gt;função&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;igualdade logarítmica&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{senh}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1325.svg\" alt=\"${\\mathrm{senh}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\ln(x + \\sqrt{x^{2} + 1})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img1326.svg\" alt=\"$\\ln(x + \\sqrt{x^{2} + 1})$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         $\\cosh^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1327.svg\" alt=\"$\\cosh^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $[1,\\infty)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\ln(x + \\sqrt{x^{2}-1})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img1328.svg\" alt=\"$\\ln(x + \\sqrt{x^{2}-1})$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm {tgh}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1329.svg\" alt=\"${\\mathrm {tgh}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{1}{2} \\ln \\left( \\tfrac{1+x}{1-x} \\right)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img1330.svg\" alt=\"$\\frac{1}{2} \\ln \\left( \\tfrac{1+x}{1-x} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{ctgh}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1331.svg\" alt=\"${\\mathrm{ctgh}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n         $\\mathbb{R}- [-1,1]$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1332.svg\" alt=\"$\\mathbb{R}- [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\frac{1}{2} \\ln \\left( \\tfrac{x+1}{x-1} \\right)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img1333.svg\" alt=\"$\\frac{1}{2} \\ln \\left( \\tfrac{x+1}{x-1} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{sech}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1334.svg\" alt=\"${\\mathrm{sech}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img887.svg\" alt=\"$(0,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\ln \\left( \\frac{1}{x} + \\frac{\\sqrt{1-x^{2}}}{x} \\right)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1335.svg\" alt=\"$\\ln \\left( \\frac{1}{x} + \\frac{\\sqrt{1-x^{2}}}{x} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n        &lt;br&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"LEFT\"&gt;&lt;!-- MATH\n         ${\\mathrm{csch}}^{-1} x$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1336.svg\" alt=\"${\\mathrm{csch}}^{-1} x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\mathbb{R}-\\{0\\}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1337.svg\" alt=\"$\\mathbb{R}-\\{0\\}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\ln \\left( \\tfrac{1}{x} + \\tfrac{\\sqrt{1 + x^{2}} }{|x|} \\right)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1338.svg\" alt=\"$\\ln \\left( \\tfrac{1}{x} + \\tfrac{\\sqrt{1 + x^{2}} }{\\vert x\\vert} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n         &lt;a id=\"tabdfhi\"&gt;&lt;/a&gt;\n        &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;/div&gt;\n        &lt;br&gt;\n        \n        &lt;p class=\" unidade\" id=\"3P186\" title=\"3P186\"&gt;\n        Note que as fórmulas de derivação das funções trigonométricas hiperbólicas inversas, foram obtidas na seção\n        &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-hiperbolicas#secderhipinv\"&gt;2.8&lt;/a&gt; e resumidas na tabela &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-hiperbolicas#tabderhipinv\"&gt;2.4&lt;/a&gt;. Naquela seção foi utilizado o método da diferenciação\n        implícita. As fórmulas de derivação da tabela &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-hiperbolicas#tabderhipinv\"&gt;2.4&lt;/a&gt; podem também ser obtidas derivando diretamente as\n        expressões logarítmicas da tabela &lt;a href=\"#tabdfhi\"&gt;3.1&lt;/a&gt;. Deixamos os detalhes para o leitor.\n        &lt;/p&gt;",
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      "Capítulo 3: Igualdades exponenciais e logarítmicas"
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    "title": "Capítulo 3: Igualdades exponenciais e logarítmicas",
    "section": "3.4 Extensão às variáveis complexas",
    "text": "3.4 Extensão às variáveis complexas\n\n            &lt;p class=\" unidade\" id=\"3P187\" title=\"3P187\"&gt;\n            Identidades similares das identidades (&lt;a href=\"#idexpo\"&gt;3.1&lt;/a&gt;) são conhecidas para as funções trigonométricas circulares. Mas\n            isto exigirá o uso de números complexos. Além disso, modelos matemáticos que representam fenômenos físicos são\n            constantemente usados para estudar e conhecer esses fenômenos — e em várias situações — a representação desses fenômenos\n            exige a utilização de números complexos juntamente com funções trigonométricas. Em virtude disso, apresentaremos nesta\n            seção como são definidas as funções trigonométricas circulares e hiperbólicas de uma variável complexa.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P188\" title=\"3P188\"&gt;\n            Usando as séries de potências das funções trigonométricas, desenvolvidas na seção anterior, vamos construir as funções\n            trigonométricas de variáveis complexas. Na seção &lt;a href=\"#secspot\"&gt;3.1&lt;/a&gt;, vimos que\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P189\" title=\"3P189\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1339.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1340.svg\" alt=\"$\\displaystyle = x - \\frac{1}{3!} x^{3} + \\frac{1}{5!} x^{5} - \\frac{1}{7!} x^{7} + \\frac{1}{9!} x^{9} - \\frac{1}{11!} x^{11} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1341.svg\" alt=\"$\\displaystyle \\cos x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1342.svg\" alt=\"$\\displaystyle = 1 - \\frac{1}{2!} x^{2} + \\frac{1}{4!} x^{4} - \\frac{1}{6!} x^{6} + \\frac{1}{8!} x^{8} - \\frac{1}{10!} x^{10} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1343.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1344.svg\" alt=\"$\\displaystyle = x + \\frac{1}{3!} x^{3} + \\frac{1}{5!} x^{5} + \\frac{1}{7!} x^{7} + \\frac{1}{9!} x^{9} + \\frac{1}{11!} x^{11} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1345.svg\" alt=\"$\\displaystyle \\cosh x$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1346.svg\" alt=\"$\\displaystyle = 1 + \\frac{1}{2!} x^{2} + \\frac{1}{4!} x^{4} + \\frac{1}{6!} x^{6} + \\frac{1}{8!} x^{8} + \\frac{1}{10!} x^{10} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P190\" style=\"text-indent: 0 !important;\" title=\"3P190\"&gt;\n            para todo &lt;!-- MATH\n             $x \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P191\" title=\"3P191\"&gt;\n            Observe que o lado direito destas igualdades faz sentido se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; for um número complexo, desde que a série seja\n            convergente para este número complexo. Isto nos sugere que a igualdade possa ser utilizada para definir as funções\n            trigonométricas seno e cosseno para os números complexos que tornam a série convergente. Nestes termos, se &lt;!-- MATH\n             $z \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n            então definimos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P192\" title=\"3P192\"&gt;&lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1348.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n           &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1349.svg\" alt=\"$\\displaystyle = z - \\frac{1}{3!} z^{3} + \\frac{1}{5!} z^{5} - \\frac{1}{7!} z^{7} + \\frac{1}{9!} z^{9} - \\frac{1}{11!} z^{11} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"senz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;12&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1350.svg\" alt=\"$\\displaystyle \\cos z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1351.svg\" alt=\"$\\displaystyle = 1 - \\frac{1}{2!} z^{2} + \\frac{1}{4!} z^{4} - \\frac{1}{6!} z^{6} + \\frac{1}{8!} z^{8} - \\frac{1}{10!} z^{10} +\n            \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"cosz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;13&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1352.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n           &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1353.svg\" alt=\"$\\displaystyle = z + \\frac{1}{3!} z^{3} + \\frac{1}{5!} z^{5} + \\frac{1}{7!} z^{7} + \\frac{1}{9!} z^{9} + \\frac{1}{11!} z^{11} + \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"senhz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;14&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1354.svg\" alt=\"$\\displaystyle \\cosh z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1355.svg\" alt=\"$\\displaystyle = 1 - \\frac{1}{2!} z^{2} + \\frac{1}{4!} z^{4} + \\frac{1}{6!} z^{6} + \\frac{1}{8!} z^{8} + \\frac{1}{10!} z^{10} +\n            \\cdots,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"coshz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;15&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P193\" style=\"text-indent: 0 !important;\" title=\"3P193\"&gt;\n            desde que as séries convirjam.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P194\" title=\"3P194\"&gt;\n            Precisamos determinar os valores &lt;!-- MATH\n             $z \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; que tornam estas séries convergentes. Para isto, recorremos ao teste da\n            razão (Critério de D'Alembert), para garantir a convergência de séries de potências de variável complexa. A\n            demonstração deste teorema pode ser encontrada em algum texto de Variáveis complexas. Recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#ZillVC\"&gt;11&lt;/a&gt;, Zill].\n            \n            &lt;/p&gt;\n            &lt;div id=\"3Teo9\" title=\"3Teo9\" class=\"unidade\"&gt;&lt;a id=\"DAlembert\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;9&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; (Teste da razão)  &nbsp; \n            &lt;i&gt;Se &lt;!-- MATH\n             $\\{z_{n}\\}_{n \\in \\mathbb{N}}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1356.svg\" alt=\"$\\{z_{n}\\}_{n \\in \\mathbb{N}}$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma sequência de números complexos e\n            &lt;/i&gt;&lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{n \\to \\infty} \\left| \\frac{z_{n+1}}{z_{n}} \\right| = L &lt; 1,\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"3P195\" title=\"3P195\"&gt;\n            &lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img1357.svg\" alt=\"$\\displaystyle \\lim_{n \\to \\infty} \\left\\vert \\frac{z_{n+1}}{z_{n}} \\right\\vert = L &lt; 1, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P196\" style=\"text-indent: 0 !important;\" title=\"3P196\"&gt;&lt;i&gt;\n            então a série &lt;!-- MATH\n             $\\sum z_{n}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1358.svg\" alt=\"$\\sum z_{n}$\" loading=\"lazy\"&gt;&lt;/span&gt; é absolutamente convergente (e, portanto, convergente).\n            &lt;/i&gt;&lt;/p&gt;&lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"3P197\" title=\"3P197\"&gt;\n            A série de potências (&lt;a href=\"#senz\"&gt;3.12&lt;/a&gt;) converge qualquer que seja &lt;!-- MATH\n             $z \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;, pois\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\lim_{n \\to \\infty} \\left| - \\frac{\\frac{1}{(2n+1)!} z^{2n+1}}{\\frac{1}{(2n-1)!} z^{2n-1}} \\right|\n              =\\lim_{n \\to \\infty} \\left| \\frac{(2n-1)!}{(2n+1)!} \\frac{z^{2n+1}}{z^{2n-1}} \\right| = \\lim_{n \\to \\infty} \\frac{1}{(2n+1)2n} |z^{2}| = 0 &lt; 1, \n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P198\" title=\"3P198\"&gt;\n            &lt;img style=\"height: 6.84ex; vertical-align: -2.91ex; \" src=\"img/img1359.svg\" alt=\"$\\displaystyle \\lim_{n \\to \\infty} \\left\\vert - \\frac{\\frac{1}{(2n+1)!} z^{2n+1}...\n            ...\\right\\vert = \\lim_{n \\to \\infty} \\frac{1}{(2n+1)2n} \\vert z^{2}\\vert = 0 &lt; 1, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P199\" style=\"text-indent: 0 !important;\" title=\"3P199\"&gt;\n            para qualquer &lt;!-- MATH\n             $z \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Analogamente para as séries de potências em (&lt;a href=\"#cosz\"&gt;3.13&lt;/a&gt;), (&lt;a href=\"#senhz\"&gt;3.14&lt;/a&gt;) e (&lt;a href=\"#coshz\"&gt;3.15&lt;/a&gt;).\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P200\" title=\"3P200\"&gt;\n            Naturalmente as definições (&lt;a href=\"#senz\"&gt;3.12&lt;/a&gt;)-(&lt;a href=\"#coshz\"&gt;3.15&lt;/a&gt;) não são muito cômodas para trabalharmos. Vamos então tentar\n            modificar estas expressões para redefinir seno e cosseno de números complexos em termos de funções reais de variável\n            real. Independente de modificarmos estas expressões, os membros na direita destas igualdades são números complexos e\n            então o que esperamos é que possamos reescrever a série de potências como sendo um número complexo mais simples de ser\n            manipulado, dado na forma tradicional &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1360.svg\" alt=\"$a + bi$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n             $a, b \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1361.svg\" alt=\"$a, b \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P201\" title=\"3P201\"&gt;\n            Comecemos então com a identidade (&lt;a href=\"#senz\"&gt;3.12&lt;/a&gt;), colocando &lt;!-- MATH\n             $z = x + yi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1362.svg\" alt=\"$z = x + yi$\" loading=\"lazy\"&gt;&lt;/span&gt;, com &lt;!-- MATH\n             $x, y \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.04ex; vertical-align: -0.50ex; \" src=\"img/img1251.svg\" alt=\"$x, y \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos então\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P202\" title=\"3P202\"&gt;&lt;a id=\"eqseno\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            {\\mathrm {sen}}(x+yi) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (x+yi)^{2n+1}.\n            \\end{equation}\n             --&gt;\n            &lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1363.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(x+yi) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (x+yi)^{2n+1}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right;  vertical-align: middle;\"&gt;\n                (&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;16&lt;/span&gt;)\n            &lt;/td&gt;&lt;/tr&gt;\n\n            &lt;/tbody&gt;&lt;/table&gt;\n\n\n\n        &lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P203\" title=\"3P203\"&gt;\n            Usando a fórmula da expansão binomial para o termo &lt;!-- MATH\n             $(x+yi)^{2n+1}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1364.svg\" alt=\"$(x+yi)^{2n+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;, podemos reescrever (&lt;a href=\"#eqseno\"&gt;3.16&lt;/a&gt;) como\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P204\" title=\"3P204\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1365.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(x+yi)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1366.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (x+yi)^{2n+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1367.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} \\sum_{r=0}^{2n+1} \\frac{(2n+1)!}{r!(2n-r+1)!} x^{2n-r+1} (yi)^{r}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1368.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{r!(2n-r+1)!} x^{2n-r+1} (yi)^{r}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P205\" title=\"3P205\"&gt;\n            O próximo lema será útil para trabalhar com o somatório duplo do segundo membro desta última igualdade.\n            \n            &lt;/p&gt;\n            &lt;div id=\"3Teo10\" title=\"3Teo10\" class=\"unidade\"&gt;&lt;b&gt;Lema  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;10&lt;/span&gt;&lt;/b&gt; &nbsp; \n            &lt;i&gt;Para qualquer &lt;!-- MATH\n             $m \\in \\mathbb{N}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1369.svg\" alt=\"$m \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n            &lt;/i&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P206\" title=\"3P206\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1370.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img1371.svg\" alt=\"$\\displaystyle \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1372.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi)^{m+1} \\cos(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1373.svg\" alt=\"$\\displaystyle \\qquad \\qquad - \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m+2)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m+2}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;/div&gt;\n            \n            \n            &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n            \n            Tomando\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m},\n            \\end{displaymath}\n             --&gt;\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"3P207\" title=\"3P207\"&gt;\n            &lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1374.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P208\" style=\"text-indent: 0 !important;\" title=\"3P208\"&gt;\n            vamos separar o caso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1375.svg\" alt=\"$n = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; do somatório externo e depois os casos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1376.svg\" alt=\"$r = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; do somatório interno. Desta forma, para\n            qualquer que seja &lt;!-- MATH\n             $m \\in \\mathbb{N}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1369.svg\" alt=\"$m \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P209\" title=\"3P209\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1377.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1378.svg\" alt=\"$\\displaystyle \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.19ex; vertical-align: -0.10ex; \" src=\"img/img1379.svg\" alt=\"$\\displaystyle %\n            $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1380.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} x (yi)^{m} + \\frac{1}{(m+1)!} (yi)^{m+1} + \\sum_{n...\n            ...infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1381.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} x (yi)^{m} + \\frac{1}{(m+1)!} (yi)^{m+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1382.svg\" alt=\"$\\displaystyle \\qquad + \\sum_{n=1}^{\\infty} \\left( \\frac{(-1)^{n}}{m!(2n+1)!} x^...\n            ...sum_{r=1}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1383.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} x (yi)^{m} + \\frac{1}{(m+1)!} (yi)^{m+1} + \\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{m!(2n+1)!} x^{2n+1} (yi)^{m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1384.svg\" alt=\"$\\displaystyle \\qquad + \\sum_{n=1}^{\\infty} \\sum_{r=1}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.67ex; vertical-align: -2.90ex; \" src=\"img/img1385.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} \\left( x + \\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} x^{2n+1} \\right) + \\frac{1}{(m+1)!} (yi)^{m+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1384.svg\" alt=\"$\\displaystyle \\qquad + \\sum_{n=1}^{\\infty} \\sum_{r=1}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1386.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi...\n            ...nfty} \\sum_{r=1}^{2n+1}\n            \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P210\" title=\"3P210\"&gt;\n            Separando novamente os temos em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1387.svg\" alt=\"$r = 1$\" loading=\"lazy\"&gt;&lt;/span&gt; do somatório interno, temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P211\" title=\"3P211\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1377.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1378.svg\" alt=\"$\\displaystyle \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.19ex; vertical-align: -0.10ex; \" src=\"img/img1379.svg\" alt=\"$\\displaystyle %\n            $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1388.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi...\n            ...infty} \\sum_{r=1}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1389.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi)^{m+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1390.svg\" alt=\"$\\displaystyle \\qquad + \\sum_{n=1}^{\\infty} \\left( \\frac{(-1)^{n}}{(m+1)!(2n)!} ...\n            ...\\sum_{r=2}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}\\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1391.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi)^{m+1} + \\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{(m+1)!(2n)!} x^{2n} (yi)^{m+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1392.svg\" alt=\"$\\displaystyle \\qquad + \\sum_{n=1}^{\\infty} \\sum_{r=2}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.67ex; vertical-align: -2.90ex; \" src=\"img/img1393.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi)^{m+1} \\left( 1 + \\sum_{n=1}^{\\infty} \\frac{(-1)^{n}}{(2n)!} x^{2n} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.89ex; vertical-align: -2.90ex; \" src=\"img/img1392.svg\" alt=\"$\\displaystyle \\qquad + \\sum_{n=1}^{\\infty} \\sum_{r=2}^{2n+1} \\frac{(-1)^{n}}{(r+m)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1372.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi)^{m+1} \\cos(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1394.svg\" alt=\"$\\displaystyle \\qquad - \\sum_{n=0}^{\\infty} \\sum_{r=2}^{2n+3} \\frac{(-1)^{n}}{(r+m)!(2n-r+3)!} x^{2n-r+3} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1372.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} {\\mathrm {sen}}(x) + \\frac{1}{(m+1)!} (yi)^{m+1} \\cos(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1395.svg\" alt=\"$\\displaystyle \\qquad - \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+m+2)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+m+2},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P212\" style=\"text-indent: 0 !important;\" title=\"3P212\"&gt;\n            exatamente como desejado.\n             &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;/p&gt;&lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"3P213\" title=\"3P213\"&gt;\n            Usando agora repetidamente este lema temos que\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P214\" title=\"3P214\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;sen&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1397.svg\" alt=\"$\\displaystyle (x+yi) = \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{r!(2n-r+1)!} x^{2n-r+1} (yi)^{r}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1398.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) + (yi)\\cos(x) - \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+2)!(2n-r+1)!} x^{2n-r+1}(yi)^{r+2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1399.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) + (yi)\\cos(x) - \\frac{1}{2!} (yi)^{2} {\\mathrm {sen}}(x) - \\frac{1}{3!} (yi)^{3} \\cos(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1400.svg\" alt=\"$\\displaystyle \\qquad \\qquad + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+4)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+4}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1401.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) + (yi)\\cos(x) - \\frac{1}{2!} (yi)^{2} {\\math...\n            ...s(x) + \\frac{1}{4!} (yi)^{4} {\\mathrm {sen}}(x) + \\frac{1}{5!} (yi)^{5} \\cos(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1402.svg\" alt=\"$\\displaystyle \\qquad \\qquad + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+6)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+6}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1401.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) + (yi)\\cos(x) - \\frac{1}{2!} (yi)^{2} {\\math...\n            ...s(x) + \\frac{1}{4!} (yi)^{4} {\\mathrm {sen}}(x) + \\frac{1}{5!} (yi)^{5} \\cos(x)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1403.svg\" alt=\"$\\displaystyle \\qquad \\qquad - \\frac{1}{6!} (yi)^{6} {\\mathrm {sen}}(x) - \\frac{...\n            ...} \\sum_{r=0}^{2n+1} \\frac{(-1)^{n}}{(r+8)!(2n-r+1)!} x^{2n-r+1} (yi)^{r+8}, %\n            $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P215\" style=\"text-indent: 0 !important;\" title=\"3P215\"&gt;\n            e assim sucessivamente. Desta forma, obtemos\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm {sen}}(x+yi) = {\\mathrm {sen}}(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n)!}(yi)^{2n}\n            + \\cos(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (yi)^{2n+1}, \n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P216\" title=\"3P216\"&gt;\n            &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1404.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(x+yi) = {\\mathrm {sen}}(x) \\sum_{n=0}^{\\infty} \\f...\n            ...(yi)^{2n}\n            + \\cos(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (yi)^{2n+1}, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P217\" style=\"text-indent: 0 !important;\" title=\"3P217\"&gt;\n            e usando o fato de que &lt;!-- MATH\n             $(yi)^{2n} = y^{2n}i^{2n} = (-1)^{n} y^{2n}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1405.svg\" alt=\"$(yi)^{2n} = y^{2n}i^{2n} = (-1)^{n} y^{2n}$\" loading=\"lazy\"&gt;&lt;/span&gt; e que &lt;!-- MATH\n             $(yi)^{2n+1} = y^{2n+1}i^{2n+1} = (-1)^{n} y^{2n+1} i$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1406.svg\" alt=\"$(yi)^{2n+1} = y^{2n+1}i^{2n+1} = (-1)^{n} y^{2n+1} i$\" loading=\"lazy\"&gt;&lt;/span&gt;, então temos que\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P218\" title=\"3P218\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1365.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(x+yi)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1407.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n)!}(yi)^{2n} + \\cos(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (yi)^{2n+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1408.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n)!}(-...\n            ...y^{2n} + \\cos(x) \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{(2n+1)!} (-1)^{n}y^{2n+1}i$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1409.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) \\sum_{n=0}^{\\infty} \\frac{1}{(2n)!}y^{2n} + i \\cos(x) \\sum_{n=0}^{\\infty} \\frac{1}{(2n+1)!}y^{2n+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1410.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(x) \\cosh(y) + i \\cos(x) {\\mathrm{senh}}(y).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P219\" title=\"3P219\"&gt;\n            Temos portanto uma definição alternativa e mais elegante para a definição do seno de um número complexo &lt;!-- MATH\n             $z = x + yi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1362.svg\" alt=\"$z = x + yi$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            Definição esta que será também útil para os nossos propósitos. Não estamos interessados em repetir o procedimento\n            anterior, mas ele pode ser aplicado também às funções cosseno, senho hiperbólico e cosseno hiperbólico para obter\n            expressões mais simples. Como não repetiremos o processo anterior apenas enunciaremos as expressões finais na próxima definição.\n            &lt;/p&gt;\n            &lt;div&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;11&lt;/span&gt;&lt;/b&gt; &nbsp; \n            Dado &lt;!-- MATH\n             $z = u + iv \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1411.svg\" alt=\"$z = u + iv \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;, os números complexos seno de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\" loading=\"lazy\"&gt;&lt;/span&gt;, cosseno de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\" loading=\"lazy\"&gt;&lt;/span&gt;, seno hiperbólico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\" loading=\"lazy\"&gt;&lt;/span&gt; e cosseno\n            hiperbólico de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\" loading=\"lazy\"&gt;&lt;/span&gt;, são dados respectivamente por,\n            \n            &lt;div class=\"mathdisplay unidade\" id=\"3P220\" title=\"3P220\"&gt;&lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1348.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1413.svg\" alt=\"$\\displaystyle = {\\mathrm {sen}}(u+iv) = {\\mathrm {sen}}u \\cosh v + i{\\mathrm{senh}}v \\cos u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"defsinz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;17&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1350.svg\" alt=\"$\\displaystyle \\cos z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1414.svg\" alt=\"$\\displaystyle = \\cos(u+iv) = \\cos u \\cosh v - i{\\mathrm {sen}}u {\\mathrm{senh}}v,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"defcosz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;18&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1352.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1415.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}(u+iv) = {\\mathrm{senh}}u \\cos v + i{\\mathrm {sen}}v \\cosh u,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            &lt;a id=\"defsenhz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;19&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center; vertical-align: middle;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1354.svg\" alt=\"$\\displaystyle \\cosh z$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1416.svg\" alt=\"$\\displaystyle = \\cosh(u+iv) = \\cosh u \\cos v + i{\\mathrm{senh}}u {\\mathrm {sen}}v.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &lt;a id=\"defcoshz\"&gt;(&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;20&lt;/span&gt;)&lt;/a&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"3P221\" title=\"3P221\"&gt;\n            Vamos analisar um pouco mais estas funções e verificar que elas possuem propriedades similares às funções\n            trigonométricas com argumentos reais. É natural esperar por isto, pois extensões não devem desorganizar o que já estava\n            “funcionando”. Comecemos com os casos circulares.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P222\" title=\"3P222\"&gt;\n            Vamos determinar as raízes das funções seno e cosseno. Queremos então determinar os valores de &lt;!-- MATH\n             $z = u + iv \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1411.svg\" alt=\"$z = u + iv \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; para\n            os quais &lt;!-- MATH\n             ${\\mathrm {sen}}z = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1417.svg\" alt=\"${\\mathrm {sen}}z = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Nestes termos queremos determinar os valores (reais) de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm {sen}}z = {\\mathrm {sen}}u \\cosh v + i {\\mathrm{senh}}v \\cos u = 0.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P223\" title=\"3P223\"&gt;\n            &lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1418.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}z = {\\mathrm {sen}}u \\cosh v + i {\\mathrm{senh}}v \\cos u = 0. $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"3P224\" title=\"3P224\"&gt;\n            Da igualdade de números complexos temos que\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P225\" title=\"3P225\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1419.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}u \\cosh v = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1420.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}v \\cos u = 0.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P226\" title=\"3P226\"&gt;\n            Da primeira equação, como &lt;!-- MATH\n             $\\cosh v \\geq 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1421.svg\" alt=\"$\\cosh v \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n             $v \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1422.svg\" alt=\"$v \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então resta que &lt;!-- MATH\n             ${\\mathrm {sen}}u = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img117.svg\" alt=\"${\\mathrm {sen}}u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos assim que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img216.svg\" alt=\"$u = k \\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Com estes valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; na segunda equação temos que &lt;!-- MATH\n             $\\cos u = \\pm 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1423.svg\" alt=\"$\\cos u = \\pm 1$\" loading=\"lazy\"&gt;&lt;/span&gt; e então resta\n            que &lt;!-- MATH\n             ${\\mathrm{senh}}v = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1424.svg\" alt=\"${\\mathrm{senh}}v = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; donde obtemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1425.svg\" alt=\"$v = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm {sen}}z = 0 \\qquad \\text{se e somente se} \\qquad z = u+iv = k\\pi,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P227\" title=\"3P227\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1426.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}z = 0$\" loading=\"lazy\"&gt;&nbsp; &nbsp;se e somente se&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1427.svg\" alt=\"$\\displaystyle \\qquad z = u+iv = k\\pi, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P228\" style=\"text-indent: 0 !important;\" title=\"3P228\"&gt;\n            para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;, exatamente como no caso real.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P229\" title=\"3P229\"&gt;\n            Analogamente para determinar os valores de &lt;!-- MATH\n             $z = u + iv \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1411.svg\" alt=\"$z = u + iv \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que &lt;!-- MATH\n             $\\cos z = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1428.svg\" alt=\"$\\cos z = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos que encontrar os valores\n            reais de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\cos z = \\cos u \\cosh v - i {\\mathrm {sen}}u {\\mathrm{senh}}v = 0,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P230\" title=\"3P230\"&gt;\n            &lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1429.svg\" alt=\"$\\displaystyle \\cos z = \\cos u \\cosh v - i {\\mathrm {sen}}u {\\mathrm{senh}}v = 0, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P231\" style=\"text-indent: 0 !important;\" title=\"3P231\"&gt;\n            e da igualdade de números complexos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P232\" title=\"3P232\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1430.svg\" alt=\"$\\displaystyle \\cos u \\cosh v = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1431.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}u {\\mathrm{senh}}v = 0.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P233\" title=\"3P233\"&gt;\n            Como &lt;!-- MATH\n             $\\cosh v \\geq 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1421.svg\" alt=\"$\\cosh v \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n             $v \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1422.svg\" alt=\"$v \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, da primeira equação resta que &lt;!-- MATH\n             $\\cos u = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img131.svg\" alt=\"$\\cos u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e então &lt;!-- MATH\n             $u = \\frac{\\pi}{2} + k\n            \\pi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img222.svg\" alt=\"$u = \\frac{\\pi}{2} + k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como &lt;!-- MATH\n             ${\\mathrm {sen}}u = \\pm 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1432.svg\" alt=\"${\\mathrm {sen}}u = \\pm 1$\" loading=\"lazy\"&gt;&lt;/span&gt;, substituindo na segunda equação vem &lt;!-- MATH\n             ${\\mathrm{senh}}v = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1424.svg\" alt=\"${\\mathrm{senh}}v = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1425.svg\" alt=\"$v = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Assim,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\cos z = 0 \\qquad \\text{se e somente se} \\qquad z = u+iv = \\frac{\\pi}{2} + k\\pi,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P234\" title=\"3P234\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1433.svg\" alt=\"$\\displaystyle \\cos z = 0$\" loading=\"lazy\"&gt;&nbsp; &nbsp;se e somente se&lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img1434.svg\" alt=\"$\\displaystyle \\qquad z = u+iv = \\frac{\\pi}{2} + k\\pi, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P235\" style=\"text-indent: 0 !important;\" title=\"3P235\"&gt;\n            para qualquer &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;, também como no caso real.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P236\" title=\"3P236\"&gt;\n            Dessa forma, as demais funções trigonométricas circulares com argumentos complexos são definidas, em termos destas duas,\n            como no caso de variável real, respeitando o domínio de definição. São portanto,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P237\" title=\"3P237\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img1435.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.06ex; vertical-align: -1.57ex; \" src=\"img/img1436.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}z}{\\cos z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img1437.svg\" alt=\"$\\displaystyle \\quad z \\neq \\frac{\\pi}{2} + k \\pi \\quad (k \\in \\mathbb{Z}),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img1438.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.06ex; vertical-align: -1.57ex; \" src=\"img/img1439.svg\" alt=\"$\\displaystyle = \\frac{\\cos z}{{\\mathrm {sen}}z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1440.svg\" alt=\"$\\displaystyle \\quad z \\neq k \\pi \\quad (k \\in \\mathbb{Z}),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1441.svg\" alt=\"$\\displaystyle \\sec z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1442.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cos z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img1437.svg\" alt=\"$\\displaystyle \\quad z \\neq \\frac{\\pi}{2} + k \\pi \\quad (k \\in \\mathbb{Z}),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1443.svg\" alt=\"$\\displaystyle \\csc z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1444.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm {sen}}z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1445.svg\" alt=\"$\\displaystyle \\quad z \\neq k \\pi \\quad (k \\in \\mathbb{Z}).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P238\" title=\"3P238\"&gt;\n            Podemos facilmente verificar, pelas igualdades (&lt;a href=\"#defsinz\"&gt;3.17&lt;/a&gt;) e (&lt;a href=\"#defcosz\"&gt;3.18&lt;/a&gt;), a validade para o caso complexo de\n            identidades conhecidas para o caso real, tais como\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P239\" title=\"3P239\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1446.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}^{2}z + \\cos^{2}z = 1,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1447.svg\" alt=\"$\\displaystyle \\cos(z+\\pi) = -\\cos z$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1448.svg\" alt=\"$\\displaystyle \\qquad {\\mathrm {sen}}(z+\\pi) = -{\\mathrm {sen}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1449.svg\" alt=\"$\\displaystyle \\cos(-z) = \\cos z$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1450.svg\" alt=\"$\\displaystyle \\qquad {\\mathrm {sen}}(-z) = -{\\mathrm {sen}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img1451.svg\" alt=\"$\\displaystyle 1+{\\mathrm {tg}}^{2}z = \\sec^{2}z$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img1452.svg\" alt=\"$\\displaystyle \\qquad 1+{\\mathrm {ctg}}^{2}z = \\csc^{2}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P240\" style=\"text-indent: 0 !important;\" title=\"3P240\"&gt;\n            dentre muitas outras.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P241\" title=\"3P241\"&gt;\n            Uma consequência direta da definição das funções seno e cosseno por série de potência é que estas funções são\n            analíticas no domínio de convergência da série, isto é, no plano complexo todo. Sendo assim, estas funções satisfazem\n            as condições de Cauchy-Riemann em todo o plano complexo e isto nos dá uma forma rápida para determinar as derivadas\n            destas duas funções. Para um estudo mais aprofundado sobre funções analíticas e as condições de Cauchy-Riemann pode-se\n            consultar algum texto sobre variáveis complexas. Recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#ZillVC\"&gt;11&lt;/a&gt;, Zill].\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P242\" title=\"3P242\"&gt;\n            Por hora basta saber que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1453.svg\" alt=\"$z = u+iv$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n             $f(z) = g(u,v)+ih(u,v)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1454.svg\" alt=\"$f(z) = g(u,v)+ih(u,v)$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma função analítica em uma região do plano\n            complexo, então\n            &lt;!-- MATH\n             \\begin{displaymath}\n            f'(z) = \\frac{\\partial g}{\\partial u} + i \\frac{\\partial h}{\\partial u} = \\frac{\\partial h}{\\partial v} - i \\frac{\\partial g}{\\partial v},\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P243\" title=\"3P243\"&gt;\n            &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1455.svg\" alt=\"$\\displaystyle f'(z) = \\frac{\\partial g}{\\partial u} + i \\frac{\\partial h}{\\partial u} = \\frac{\\partial h}{\\partial v} - i \\frac{\\partial g}{\\partial v}, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P244\" style=\"text-indent: 0 !important;\" title=\"3P244\"&gt;\n            para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\" loading=\"lazy\"&gt;&lt;/span&gt; nesta região.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P245\" title=\"3P245\"&gt;\n            Considerando &lt;!-- MATH\n             $f(z) = {\\mathrm {sen}}z = g(u,v) + i h(u,v)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1456.svg\" alt=\"$f(z) = {\\mathrm {sen}}z = g(u,v) + i h(u,v)$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos &lt;!-- MATH\n             $g(u,v) = {\\mathrm {sen}}u \\cosh v$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1457.svg\" alt=\"$g(u,v) = {\\mathrm {sen}}u \\cosh v$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n             $h(u,v) = {\\mathrm{senh}}v \\cos u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1458.svg\" alt=\"$h(u,v) = {\\mathrm{senh}}v \\cos u$\" loading=\"lazy\"&gt;&lt;/span&gt; e então\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P246\" title=\"3P246\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1459.svg\" alt=\"$\\displaystyle f'(z) = ({\\mathrm {sen}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1460.svg\" alt=\"$\\displaystyle = \\frac{\\partial g}{\\partial u} + i \\frac{\\partial h}{\\partial u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1461.svg\" alt=\"$\\displaystyle = \\frac{\\partial}{\\partial u} ({\\mathrm {sen}}u \\cosh v) + i \\frac{\\partial}{\\partial u}({\\mathrm{senh}}v \\cos u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1462.svg\" alt=\"$\\displaystyle = \\cos u \\cosh v - i {\\mathrm{senh}}v {\\mathrm {sen}}u = \\cos z.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P247\" title=\"3P247\"&gt;\n            Também, se &lt;!-- MATH\n             $f(z) = \\cos z = g(u,v) + i h(u,v)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1463.svg\" alt=\"$f(z) = \\cos z = g(u,v) + i h(u,v)$\" loading=\"lazy\"&gt;&lt;/span&gt; então &lt;!-- MATH\n             $g(u,v) = \\cos u \\cosh v$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1464.svg\" alt=\"$g(u,v) = \\cos u \\cosh v$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n             $h(u,v) = -{\\mathrm {sen}}u {\\mathrm{senh}}v$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1465.svg\" alt=\"$h(u,v) = -{\\mathrm {sen}}u {\\mathrm{senh}}v$\" loading=\"lazy\"&gt;&lt;/span&gt; e, dessa\n            forma,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P248\" title=\"3P248\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1466.svg\" alt=\"$\\displaystyle f'(z) = (\\cos z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1460.svg\" alt=\"$\\displaystyle = \\frac{\\partial g}{\\partial u} + i \\frac{\\partial h}{\\partial u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1467.svg\" alt=\"$\\displaystyle = \\frac{\\partial}{\\partial u} (\\cos u \\cosh v) + i \\frac{\\partial}{\\partial u}(- {\\mathrm {sen}}u {\\mathrm{senh}}v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1468.svg\" alt=\"$\\displaystyle = -{\\mathrm {sen}}u \\cosh v - i {\\mathrm{senh}}v \\cos u = - {\\mathrm {sen}}z.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P249\" title=\"3P249\"&gt;\n            As derivadas das funções seno e cosseno de variável complexa são então respectivamente o cosseno e o oposto do seno,\n            exatamente como no caso real. Já que as regras de derivação para funções complexas são as mesmas para funções reais,\n            isto é,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P250\" title=\"3P250\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1469.svg\" alt=\"$\\displaystyle (f(z)+g(z))'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1470.svg\" alt=\"$\\displaystyle = f'(z) + g'(z),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1471.svg\" alt=\"$\\displaystyle (f(z) \\cdot g(z))'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1472.svg\" alt=\"$\\displaystyle = f'(z) \\cdot g(z) + f(z) \\cdot g'(z),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.49ex; vertical-align: -2.10ex; \" src=\"img/img1473.svg\" alt=\"$\\displaystyle \\left( \\frac{f(z)}{g(z)} \\right)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.20ex; vertical-align: -2.07ex; \" src=\"img/img1474.svg\" alt=\"$\\displaystyle = \\frac{f'(z) \\cdot g(z) - f(z) \\cdot g'(z)}{(g(z))^{2}},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P251\" style=\"text-indent: 0 !important;\" title=\"3P251\"&gt;\n            então as derivadas das demais funções trigonométricas circulares, são também iguais às derivadas obtidas no caso real.\n            São portanto\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P252\" title=\"3P252\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1475.svg\" alt=\"$\\displaystyle ({\\mathrm {tg}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img1476.svg\" alt=\"$\\displaystyle = \\sec^{2}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1477.svg\" alt=\"$\\displaystyle ({\\mathrm {ctg}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1478.svg\" alt=\"$\\displaystyle = -\\csc^{2}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1479.svg\" alt=\"$\\displaystyle (\\sec z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img1480.svg\" alt=\"$\\displaystyle = \\sec z {\\mathrm {tg}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1481.svg\" alt=\"$\\displaystyle (\\csc z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.90ex; vertical-align: -0.58ex; \" src=\"img/img1482.svg\" alt=\"$\\displaystyle = -\\csc z {\\mathrm {ctg}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P253\" style=\"text-indent: 0 !important;\" title=\"3P253\"&gt;\n            respeitados os domínios de definição.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P254\" title=\"3P254\"&gt;\n            Além disso, as identidades obtidas &lt;!-- MATH\n             $\\cos(iu) = \\cosh u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1483.svg\" alt=\"$\\cos(iu) = \\cosh u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n             ${\\mathrm {sen}}(iu) = i {\\mathrm{senh}}u$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1484.svg\" alt=\"${\\mathrm {sen}}(iu) = i {\\mathrm{senh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt; permitem estabelecer uma\n            correspondência entre as funções trigonométricas circulares e as suas respectivas hiperbólicas. As correspondências das\n            demais funções trigonométricas ficam\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P255\" title=\"3P255\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1485.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}(ui)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img1486.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}(ui)}{\\cos(ui)} = \\frac{i{\\mathrm{senh}}u}{\\cosh u} = i{\\mathrm {tgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1487.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}(ui)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img1488.svg\" alt=\"$\\displaystyle = \\frac{\\cos(ui)}{{\\mathrm {sen}}(ui)} = \\frac{\\cosh u}{i{\\mathrm{senh}}u} = \\frac{-i\\cosh u}{{\\mathrm{senh}}u} = -i{\\mathrm{ctgh}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1489.svg\" alt=\"$\\displaystyle \\sec(ui)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1490.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cos(ui)} = \\frac{1}{\\cosh u} = {\\mathrm{sech}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1491.svg\" alt=\"$\\displaystyle \\csc(ui)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1492.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm {sen}}(ui)} = \\frac{1}{i{\\mathrm{senh}}u} = \\frac{-i}{{\\mathrm{senh}}u} = -i {\\mathrm{csch}}u.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P256\" title=\"3P256\"&gt;\n            Vamos estudar outras propriedades das funções trigonométricas hiperbólicas de variável complexa. Comecemos pelas raízes\n            destas funções. Queremos determinar os números complexos &lt;!-- MATH\n             $z = u+iv \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1411.svg\" alt=\"$z = u + iv \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que &lt;!-- MATH\n             ${\\mathrm{senh}}z = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1493.svg\" alt=\"${\\mathrm{senh}}z = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Nestes termos\n            devemos encontrar números reais &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}z = {\\mathrm{senh}}u \\cos v + i{\\mathrm {sen}}v \\cosh u = 0,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P257\" title=\"3P257\"&gt;\n            &lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1494.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}z = {\\mathrm{senh}}u \\cos v + i{\\mathrm {sen}}v \\cosh u = 0, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P258\" style=\"text-indent: 0 !important;\" title=\"3P258\"&gt;\n            e da igualdade de complexos, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; devem satisfazer\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P259\" title=\"3P259\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1495.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u \\cos v = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1496.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}v \\cosh u = 0.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P260\" title=\"3P260\"&gt;\n            Da segunda equação, como &lt;!-- MATH\n             $\\cosh u \\geq 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1497.svg\" alt=\"$\\cosh u \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n             $u \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, então devemos ter &lt;!-- MATH\n             ${\\mathrm {sen}}v = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1498.svg\" alt=\"${\\mathrm {sen}}v = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1499.svg\" alt=\"$v = k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt;\n            para &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Com estes valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; na primeira equação, resta que &lt;!-- MATH\n             ${\\mathrm{senh}}u = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1500.svg\" alt=\"${\\mathrm{senh}}u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos assim que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm{senh}}z = 0 \\qquad \\text{se e somente se} \\qquad z = ik\\pi,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P261\" title=\"3P261\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1501.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}z = 0$\" loading=\"lazy\"&gt;&nbsp; &nbsp;se e somente se&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1502.svg\" alt=\"$\\displaystyle \\qquad z = ik\\pi, $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P262\" style=\"text-indent: 0 !important;\" title=\"3P262\"&gt;\n            para &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Observe que estas raízes são complexas e que a única destas raízes que é real, é &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1503.svg\" alt=\"$z = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, que coincide\n            com a única raiz real da função seno hiperbólico a argumento real.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P263\" title=\"3P263\"&gt;\n            Agora vamos determinar &lt;!-- MATH\n             $z = u+iv \\in \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1411.svg\" alt=\"$z = u + iv \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que &lt;!-- MATH\n             $\\cosh z = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1504.svg\" alt=\"$\\cosh z = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Da identidade (&lt;a href=\"#defcoshz\"&gt;3.20&lt;/a&gt;), queremos determinar os\n            valores reais de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; que satisfazem\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P264\" title=\"3P264\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1505.svg\" alt=\"$\\displaystyle \\cosh u \\cos v = 0,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1506.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}u {\\mathrm {sen}}v = 0.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P265\" title=\"3P265\"&gt;\n            Como &lt;!-- MATH\n             $\\cosh u \\geq 1$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1497.svg\" alt=\"$\\cosh u \\geq 1$\" loading=\"lazy\"&gt;&lt;/span&gt; então da primeira equação segue que &lt;!-- MATH\n             $\\cos v = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1507.svg\" alt=\"$\\cos v = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e, portanto, &lt;!-- MATH\n             $v = \\frac{\\pi}{2} + k\\pi$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img1508.svg\" alt=\"$v = \\frac{\\pi}{2} + k\\pi$\" loading=\"lazy\"&gt;&lt;/span&gt; para &lt;!-- MATH\n             $k\n            \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Com estes valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\" loading=\"lazy\"&gt;&lt;/span&gt; na segunda equação temos que &lt;!-- MATH\n             ${\\mathrm{senh}}u = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1500.svg\" alt=\"${\\mathrm{senh}}u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; e então &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img239.svg\" alt=\"$u = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;. Segue que\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\cosh z = 0 \\qquad \\text{se e somente se} \\qquad z = i(\\tfrac{\\pi}{2}+k\\pi),\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P266\" title=\"3P266\"&gt;\n            &lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1509.svg\" alt=\"$\\displaystyle \\cosh z = 0$\" loading=\"lazy\"&gt;&nbsp; &nbsp;se e somente se&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1510.svg\" alt=\"$\\displaystyle \\qquad z = i(\\tfrac{\\pi}{2}+k\\pi), $\" loading=\"lazy\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P267\" style=\"text-indent: 0 !important;\" title=\"3P267\"&gt;\n            para &lt;!-- MATH\n             $k \\in \\mathbb{Z}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Dentre estas raízes complexas não existe nenhuma raiz real, o que ratifica a não existência de números\n            reais &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; tais que &lt;!-- MATH\n             $\\cosh x = 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1511.svg\" alt=\"$\\cosh x = 0$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P268\" title=\"3P268\"&gt;\n            As demais funções trigonométricas hiperbólicas de variáveis complexas são definidas em termos de seno e cosseno como no\n            caso real, restritas ao domínio de definição. Isto é,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P269\" title=\"3P269\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1512.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1513.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm{senh}}z}{\\cosh z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1514.svg\" alt=\"$\\displaystyle \\quad z \\neq i(\\tfrac{\\pi}{2}+k\\pi) \\quad (k \\in \\mathbb{Z}),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1515.svg\" alt=\"$\\displaystyle {\\mathrm{ctgh}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1516.svg\" alt=\"$\\displaystyle = \\frac{\\cosh z}{{\\mathrm{senh}}z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1517.svg\" alt=\"$\\displaystyle \\quad z \\neq ik\\pi \\quad (k \\in \\mathbb{Z}),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1518.svg\" alt=\"$\\displaystyle {\\mathrm{sech}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1519.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cosh z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1514.svg\" alt=\"$\\displaystyle \\quad z \\neq i(\\tfrac{\\pi}{2}+k\\pi) \\quad (k \\in \\mathbb{Z}),$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1520.svg\" alt=\"$\\displaystyle {\\mathrm{csch}}z$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1521.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm{senh}}z}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;para todo&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1522.svg\" alt=\"$\\displaystyle \\quad z \\neq ik\\pi \\quad (k \\in \\mathbb{Z}).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"3P270\" title=\"3P270\"&gt;\n            As definições de seno e cosseno hiperbólicos em termos de séries de potências, convergentes em todo o plano complexo,\n            nos diz que estas funções são analíticas em todo o plano complexo e então podemos determinar facilmente as derivadas\n            destas funções. Considerando que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1453.svg\" alt=\"$z = u+iv$\" loading=\"lazy\"&gt;&lt;/span&gt; e que &lt;!-- MATH\n             $f(z) = {\\mathrm{senh}}z = g(u,v) + i h(u,v)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1523.svg\" alt=\"$f(z) = {\\mathrm{senh}}z = g(u,v) + i h(u,v)$\" loading=\"lazy\"&gt;&lt;/span&gt; temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P271\" title=\"3P271\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1524.svg\" alt=\"$\\displaystyle f'(z) = ({\\mathrm{senh}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1460.svg\" alt=\"$\\displaystyle = \\frac{\\partial g}{\\partial u} + i \\frac{\\partial h}{\\partial u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1525.svg\" alt=\"$\\displaystyle = \\frac{\\partial}{\\partial u}({\\mathrm{senh}}u \\cos v) + i\\frac{\\partial}{\\partial u}({\\mathrm {sen}}v \\cosh u)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1526.svg\" alt=\"$\\displaystyle = \\cosh u \\cos v + i{\\mathrm {sen}}v {\\mathrm{senh}}u = \\cosh z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P272\" style=\"text-indent: 0 !important;\" title=\"3P272\"&gt;\n            exatamente como no caso de variáveis reais.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P273\" title=\"3P273\"&gt;\n            Analogamente, para a função &lt;!-- MATH\n             $f(z) = \\cosh z = g(u,v) + ih(u,v)$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1527.svg\" alt=\"$f(z) = \\cosh z = g(u,v) + ih(u,v)$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P274\" title=\"3P274\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1528.svg\" alt=\"$\\displaystyle f'(z) = (\\cosh z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1460.svg\" alt=\"$\\displaystyle = \\frac{\\partial g}{\\partial u} + i \\frac{\\partial h}{\\partial u}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1529.svg\" alt=\"$\\displaystyle = \\frac{\\partial}{\\partial u}(\\cosh u \\cos v) + i\\frac{\\partial}{\\partial u}({\\mathrm{senh}}u {\\mathrm {sen}}v)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1530.svg\" alt=\"$\\displaystyle = {\\mathrm{senh}}u \\cos v + i\\cosh u {\\mathrm {sen}}v = {\\mathrm{senh}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P275\" style=\"text-indent: 0 !important;\" title=\"3P275\"&gt;\n            também como no caso real. Considerando ainda que a regra de derivação para o quociente de funções de variáveis\n            complexas é idêntica à regra de derivação para o quociente de funções de variáveis reais, então temos que\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"3P276\" title=\"3P276\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1531.svg\" alt=\"$\\displaystyle ({\\mathrm {tgh}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.16ex; vertical-align: -0.13ex; \" src=\"img/img1532.svg\" alt=\"$\\displaystyle = {\\mathrm{sech}}^{2} z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1533.svg\" alt=\"$\\displaystyle ({\\mathrm{ctgh}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1534.svg\" alt=\"$\\displaystyle = -{\\mathrm{csch}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1535.svg\" alt=\"$\\displaystyle ({\\mathrm{sech}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1536.svg\" alt=\"$\\displaystyle = -{\\mathrm{sech}}z {\\mathrm {tgh}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1537.svg\" alt=\"$\\displaystyle ({\\mathrm{csch}}z)'$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img1538.svg\" alt=\"$\\displaystyle = -{\\mathrm{csch}}z {\\mathrm{ctgh}}z,$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"3P277\" style=\"text-indent: 0 !important;\" title=\"3P277\"&gt;\n            com a devida restrição do domínio de definição. São as mesmas fórmulas de derivação que as funções trigonométricas\n            hiperbólicas de variáveis reais.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"3P278\" title=\"3P278\"&gt;\n            Podem ainda ser definidas as funções trigonométricas circulares e hiperbólicas inversas a argumentos complexos. Não\n            vamos nos estender neste aspecto, em virtude de que o caso complexo não é o foco do nosso interesse. Além disso,\n            entraríamos no campo das funções multivalentes, isto é, funções &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1539.svg\" alt=\"$f(z)$\" loading=\"lazy\"&gt;&lt;/span&gt; que assumem mais de um valor para cada &lt;!-- MATH\n             $z \\in\n            \\mathbb{C}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Esta categoria de funções foge do conceito de função de um curso de Cálculo Diferencial e Integral.\n            &lt;/p&gt;",
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    "title": "Capítulo 3: Igualdades exponenciais e logarítmicas",
    "section": "3.5 Fórmulas exponenciais para funções trigonométricas circulares",
    "text": "3.5 Fórmulas exponenciais para funções trigonométricas circulares\n\n                &lt;p class=\" unidade\" id=\"3P279\" title=\"3P279\"&gt;\n                Nesta seção, obteremos fórmulas exponenciais similares às identidades em (&lt;a href=\"#idexpo\"&gt;3.1&lt;/a&gt;) para as funções trigonométricas\n                circulares. Mais precisamente, provaremos que\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P280\" title=\"3P280\"&gt;&lt;a id=\"idexpocirc\"&gt;&lt;/a&gt;&lt;!-- MATH\n                 \\begin{equation}\n                {\\mathrm {sen}}x = \\frac{e^{ix}-e^{-ix}}{2i}  \\qquad \\text{e} \\qquad \\cos x = \\frac{e^{ix} + e^{-ix}}{2}.\n                \\end{equation}\n                 --&gt;\n                &lt;table&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1540.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}x = \\frac{e^{ix}-e^{-ix}}{2i}$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1541.svg\" alt=\"$\\displaystyle \\qquad \\cos x = \\frac{e^{ix} + e^{-ix}}{2}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                (&lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;21&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"3P281\" title=\"3P281\"&gt;\n                Observe que o lado direito destas igualdades envolve a função exponencial de variável complexa. Precisamos definir a\n                função exponencial de variável complexa e o faremos como na seção anterior onde definimos as funções trigonométricas\n                de variáveis complexas. A expansão em série de potências da função &lt;!-- MATH\n                 $f(x) = e^{x}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1147.svg\" alt=\"$f(x) = e^{x}$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n                 $x \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, é\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{x} = 1 + x + \\frac{1}{2!} x^{2} + \\frac{1}{3!} x^{3} + \\frac{1}{4!} x^{4} + \\frac{1}{5!} x^{5} + \\frac{1}{6!} x^{6} + \\cdots.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P282\" title=\"3P282\"&gt;\n                &lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1542.svg\" alt=\"$\\displaystyle e^{x} = 1 + x + \\frac{1}{2!} x^{2} + \\frac{1}{3!} x^{3} + \\frac{1}{4!} x^{4} + \\frac{1}{5!} x^{5} + \\frac{1}{6!} x^{6} + \\cdots. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"3P283\" title=\"3P283\"&gt;\n                Mas a série de potências do lado direito da igualdade faz sentido se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\" loading=\"lazy\"&gt;&lt;/span&gt; for um número complexo que torne a série\n                convergente. Definimos então por esta série de potências a função exponencial de variável complexa dada por\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{z} = 1 + z + \\frac{1}{2!} z^{2} + \\frac{1}{3!} z^{3} + \\frac{1}{4!} z^{4} + \\frac{1}{5!} z^{5} + \\frac{1}{6!} z^{6} + \\cdots,\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P284\" title=\"3P284\"&gt;\n                &lt;img style=\"height: 4.56ex; vertical-align: -1.59ex; \" src=\"img/img1543.svg\" alt=\"$\\displaystyle e^{z} = 1 + z + \\frac{1}{2!} z^{2} + \\frac{1}{3!} z^{3} + \\frac{1}{4!} z^{4} + \\frac{1}{5!} z^{5} + \\frac{1}{6!} z^{6} + \\cdots, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P285\" style=\"text-indent: 0 !important;\" title=\"3P285\"&gt;\n                para todo &lt;!-- MATH\n                 $z \\in \\mathbb{C}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; tal que a série seja convergente.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"3P286\" title=\"3P286\"&gt;\n                De acordo com o teste da razão (Teorema &lt;a href=\"#DAlembert\"&gt;3.9&lt;/a&gt;), esta série de potências é convergente em todo o plano\n                complexo, já que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\lim_{n \\to \\infty} \\left| \\frac{\\frac{1}{(n+1)!}z^{n+1}}{\\frac{1}{n!}z^{n}} \\right|\n                  = \\lim_{n \\to \\infty} \\left| \\frac{n!}{(n+1)!} \\frac{z^{n+1}}{z^{n}} \\right| = \\lim_{n \\to \\infty} \\frac{1}{(n+1)} |z| = 0 &lt; 1, \n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P287\" title=\"3P287\"&gt;\n                &lt;img style=\"height: 6.68ex; vertical-align: -2.74ex; \" src=\"img/img1544.svg\" alt=\"$\\displaystyle \\lim_{n \\to \\infty} \\left\\vert \\frac{\\frac{1}{(n+1)!}z^{n+1}}{\\fr...\n                ...z^{n}} \\right\\vert = \\lim_{n \\to \\infty} \\frac{1}{(n+1)} \\vert z\\vert = 0 &lt; 1, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P288\" style=\"text-indent: 0 !important;\" title=\"3P288\"&gt;\n                para qualquer &lt;!-- MATH\n                 $z \\in \\mathbb{C}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1347.svg\" alt=\"$z \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"3P289\" title=\"3P289\"&gt;\n                Esta série é importante, porém, dificulta o trabalho com a função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1545.svg\" alt=\"$e^{z}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Como de costume, vamos reescrever esta série\n                em termos mais agradáveis. Mais precisamente, já que o lado direito da série de potências é um número complexo,\n                esperamos poder escrever este número complexo na tradicional forma algébrica &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1360.svg\" alt=\"$a + bi$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n                 $a,b \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1361.svg\" alt=\"$a, b \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"3P290\" title=\"3P290\"&gt;\n                Tomando então &lt;!-- MATH\n                 $z = x + yi$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1362.svg\" alt=\"$z = x + yi$\" loading=\"lazy\"&gt;&lt;/span&gt; com &lt;!-- MATH\n                 $x, y \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.04ex; vertical-align: -0.50ex; \" src=\"img/img1251.svg\" alt=\"$x, y \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, aplicando a expansão binomial, podemos reescrever a função\n                exponencial na forma\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P291\" title=\"3P291\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.10ex; \" src=\"img/img1546.svg\" alt=\"$\\displaystyle e^{x + yi}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1547.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\frac{1}{n!} (x+yi)^{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1548.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\frac{1}{n!} \\sum_{r=0}^{n} \\frac{n!}{r!(n-...\n                ...)^{r}\n                = \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{r!(n-r)!} x^{n-r} (yi)^{r}.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"3P292\" title=\"3P292\"&gt;\n                O lema a seguir nos ajudará a trabalhar com o somatório duplo do segundo membro desta última igualdade.\n                \n                &lt;/p&gt;\n                &lt;div id=\"3Teo12\" title=\"3Teo12\" class=\"unidade\"&gt;&lt;b&gt;Lema  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;12&lt;/span&gt;&lt;/b&gt; &nbsp; \n                &lt;i&gt;Para qualquer &lt;!-- MATH\n                 $m \\in \\mathbb{N}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1369.svg\" alt=\"$m \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\n                &lt;/i&gt;&lt;!-- MATH\n                 \\begin{displaymath}\n                \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m}\n                 = e^{x}\\frac{1}{m!}(yi)^{m} + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m+1)!(n-r)!} x^{n-r} (yi)^{r+m+1}. \n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"3P293\" title=\"3P293\"&gt;\n                &lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1549.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r}...\n                ...m_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m+1)!(n-r)!} x^{n-r} (yi)^{r+m+1}. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                \n                &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                \n                Dado qualquer &lt;!-- MATH\n                 $m \\in \\mathbb{N}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1369.svg\" alt=\"$m \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt; e começando com\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m},\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"3P294\" title=\"3P294\"&gt;\n                &lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1550.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m}, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P295\" style=\"text-indent: 0 !important;\" title=\"3P295\"&gt;\n                vamos separar o caso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1375.svg\" alt=\"$n = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; do somatório externo e depois os casos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1376.svg\" alt=\"$r = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; do somatório interno. Desta forma, para\n                qualquer que seja &lt;!-- MATH\n                 $m \\in \\mathbb{N}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1369.svg\" alt=\"$m \\in \\mathbb{N}$\" loading=\"lazy\"&gt;&lt;/span&gt;, obtemos\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P296\" title=\"3P296\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1551.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img1552.svg\" alt=\"$\\displaystyle \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 0.19ex; vertical-align: -0.10ex; \" src=\"img/img1379.svg\" alt=\"$\\displaystyle %\n                $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1553.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} x^{0} (yi)^{m} + \\sum_{n=1}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.67ex; vertical-align: -2.90ex; \" src=\"img/img1554.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} + \\sum_{n=1}^{\\infty} \\left( \\frac{1}{m!n...\n                ...{n} (yi)^{m}\n                + \\sum_{r=1}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.53ex; vertical-align: -2.90ex; \" src=\"img/img1555.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} + \\sum_{n=1}^{\\infty} \\frac{1}{m!n!} x^{n...\n                ...\n                + \\sum_{n=1}^{\\infty} \\sum_{r=1}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.67ex; vertical-align: -2.90ex; \" src=\"img/img1556.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} \\left( 1 + \\sum_{n=1}^{\\infty} \\frac{1}{n...\n                ...\n                + \\sum_{n=1}^{\\infty} \\sum_{r=1}^{n} \\frac{1}{(r+m)!(n-r)!} x^{n-r} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.91ex; vertical-align: -2.91ex; \" src=\"img/img1557.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} e^{x} + \\sum_{n=0}^{\\infty} \\sum_{r=1}^{n+1} \\frac{1}{(r+m)!(n+1-r)!} x^{n+1-r} (yi)^{r+m}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1558.svg\" alt=\"$\\displaystyle = \\frac{1}{m!} (yi)^{m} e^{x} + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+m+1)!(n-r)!} x^{n-r} (yi)^{r+m+1},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                &lt;p class=\" unidade\" id=\"3P297\" style=\"text-indent: 0 !important;\" title=\"3P297\"&gt;\n                como desejado.\n                &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\" loading=\"lazy\"&gt;\n                 &lt;/span&gt;&lt;/p&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"3P298\" title=\"3P298\"&gt;\n                Usando agora repetidamente este lema, temos que\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P299\" title=\"3P299\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.10ex; \" src=\"img/img1546.svg\" alt=\"$\\displaystyle e^{x + yi}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1559.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{r!(n-r)!} x^{n-r} (yi)^{r}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1560.svg\" alt=\"$\\displaystyle = e^{x} + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+1)!(n-r)!} x^{n-r} (yi)^{r+1}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1561.svg\" alt=\"$\\displaystyle = e^{x} + e^{x} (yi) + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+2)!(n-r)!} x^{n-r} (yi)^{r+2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1562.svg\" alt=\"$\\displaystyle = e^{x} + e^{x} (yi) + e^{x} \\frac{1}{2!} (yi)^{2} + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+3)!(n-r)!} x^{n-r} (yi)^{r+3}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1563.svg\" alt=\"$\\displaystyle = e^{x} + e^{x} (yi) + e^{x} \\frac{1}{2!} (yi)^{2} + e^{x} \\frac{...\n                ...\n                + \\sum_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+4)!(n-r)!} x^{n-r} (yi)^{r+4}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.55ex; vertical-align: -2.91ex; \" src=\"img/img1564.svg\" alt=\"$\\displaystyle = e^{x} + e^{x} (yi) + e^{x} \\frac{1}{2!} (yi)^{2} + e^{x} \\frac{...\n                ...um_{n=0}^{\\infty} \\sum_{r=0}^{n} \\frac{1}{(r+5)!(n-r)!} x^{n-r} (yi)^{r+5}, %\n                $\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                &lt;p class=\" unidade\" id=\"3P300\" style=\"text-indent: 0 !important;\" title=\"3P300\"&gt;\n                e assim sucessivamente. Desta forma, obtemos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{x+yi} = \\sum_{n=0}^{\\infty} e^{x} \\frac{1}{n!} (yi)^{n} = e^{x} \\sum_{n=0}^{\\infty} \\frac{1}{n!} (yi)^{n},\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P301\" title=\"3P301\"&gt;\n                &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1565.svg\" alt=\"$\\displaystyle e^{x+yi} = \\sum_{n=0}^{\\infty} e^{x} \\frac{1}{n!} (yi)^{n} = e^{x} \\sum_{n=0}^{\\infty} \\frac{1}{n!} (yi)^{n}, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P302\" style=\"text-indent: 0 !important;\" title=\"3P302\"&gt;\n                e usando o fato de que quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1566.svg\" alt=\"$n = 2r$\" loading=\"lazy\"&gt;&lt;/span&gt; é par, temos que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                (yi)^{n} = (yi)^{2r} = y^{2r} i^{2r} = (-1)^{r} y^{2r},\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P303\" title=\"3P303\"&gt;\n                &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1567.svg\" alt=\"$\\displaystyle (yi)^{n} = (yi)^{2r} = y^{2r} i^{2r} = (-1)^{r} y^{2r}, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P304\" style=\"text-indent: 0 !important;\" title=\"3P304\"&gt;\n                e quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1568.svg\" alt=\"$n = 2r+1$\" loading=\"lazy\"&gt;&lt;/span&gt; é impar,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                (yi)^{n} = (yi)^{2r+1} = y^{2r+1} i^{2r+1} = (-1)^{r} y^{2r+1}i,\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P305\" title=\"3P305\"&gt;\n                &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1569.svg\" alt=\"$\\displaystyle (yi)^{n} = (yi)^{2r+1} = y^{2r+1} i^{2r+1} = (-1)^{r} y^{2r+1}i, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P306\" style=\"text-indent: 0 !important;\" title=\"3P306\"&gt;\n                então podemos separar o último somatório nas suas parcelas com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; par e com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\" loading=\"lazy\"&gt;&lt;/span&gt; ímpar e obtemos\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P307\" title=\"3P307\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.10ex; \" src=\"img/img1546.svg\" alt=\"$\\displaystyle e^{x + yi}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1570.svg\" alt=\"$\\displaystyle = e^{x} \\sum_{n=0}^{\\infty} \\frac{1}{n!} (yi)^{n}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.68ex; vertical-align: -2.91ex; \" src=\"img/img1571.svg\" alt=\"$\\displaystyle = e^{x} \\left( \\sum_{n=0}^{\\infty} \\frac{1}{(2n)!} (yi)^{2n} + \\sum_{n=0}^{\\infty} \\frac{1}{(2n+1)!} (yi)^{2n+1} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.68ex; vertical-align: -2.91ex; \" src=\"img/img1572.svg\" alt=\"$\\displaystyle = e^{x} \\left( \\sum_{n=0}^{\\infty} \\frac{1}{(2n)!}(-1)^{n}y^{2n} + \\sum_{n=0}^{\\infty} \\frac{1}{(2n+1)!}(-1)^{n}y^{2n+1}i \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.68ex; vertical-align: -2.91ex; \" src=\"img/img1573.svg\" alt=\"$\\displaystyle = e^{x} \\left( \\sum_{n=0}^{\\infty} \\frac{1}{(2n)!}(-1)^{n}y^{2n} + i\\sum_{n=0}^{\\infty} \\frac{1}{(2n+1)!}(-1)^{n}y^{2n+1} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1574.svg\" alt=\"$\\displaystyle = e^{x} \\left( \\cos(y) + i{\\mathrm {sen}}(y) \\right).$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"3P308\" title=\"3P308\"&gt;\n                Formalmente, temos então uma definição alternativa para a exponencial de um número complexo sem o uso explícito das\n                séries de potência.\n                &lt;/p&gt;\n                &lt;div&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;3&lt;/span&gt;.&lt;span class=\"arabic\"&gt;13&lt;/span&gt;&lt;/b&gt; &nbsp; \n                Dado &lt;!-- MATH\n                 $z = x + yi \\in \\mathbb{C}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1575.svg\" alt=\"$z = x + yi \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt;, definimos a exponencial de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\" loading=\"lazy\"&gt;&lt;/span&gt;, como sendo o número complexo representado por\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1576.svg\" alt=\"$e^{x+yi}$\" loading=\"lazy\"&gt;&lt;/span&gt; e dado por\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{x+yi} = e^{x} \\left( \\cos(y) + i{\\mathrm {sen}}(y) \\right) = e^{x}\\cos(y) + ie^{x}{\\mathrm {sen}}(y).\n                \\end{displaymath}\n                 --&gt;\n                \n                &lt;div class=\"mathdisplay unidade\" id=\"3P309\" title=\"3P309\"&gt;\n                &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1577.svg\" alt=\"$\\displaystyle e^{x+yi} = e^{x} \\left( \\cos(y) + i{\\mathrm {sen}}(y) \\right) = e^{x}\\cos(y) + ie^{x}{\\mathrm {sen}}(y). $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"3P310\" title=\"3P310\"&gt;\n                Agora estamos prontos para obter as identidades em (&lt;a href=\"#idexpocirc\"&gt;3.21&lt;/a&gt;). Dado &lt;!-- MATH\n                 $u \\in \\mathbb{R}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos desta última definição\n                que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{iu} = \\cos u + i {\\mathrm {sen}}u,\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P311\" title=\"3P311\"&gt;\n                &lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1578.svg\" alt=\"$\\displaystyle e^{iu} = \\cos u + i {\\mathrm {sen}}u, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P312\" title=\"3P312\"&gt;\n                e também\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{-iu} = \\cos(-u) + i{\\mathrm {sen}}(-u) = \\cos u - i{\\mathrm {sen}}u.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P313\" style=\"text-indent: 0 !important;\" title=\"3P313\"&gt;\n                &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1579.svg\" alt=\"$\\displaystyle e^{-iu} = \\cos(-u) + i{\\mathrm {sen}}(-u) = \\cos u - i{\\mathrm {sen}}u. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"3P314\" title=\"3P314\"&gt;\n                Somando estas duas últimas igualdades, temos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{iu} + e^{-iu} = 2\\cos u,\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P315\" title=\"3P315\"&gt;\n                &lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1580.svg\" alt=\"$\\displaystyle e^{iu} + e^{-iu} = 2\\cos u, $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"3P316\" style=\"text-indent: 0 !important;\" title=\"3P316\"&gt;\n                e subtraindo a segunda da primeira, temos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                e^{iu} - e^{-iu} = 2i{\\mathrm {sen}}u.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P317\" title=\"3P317\"&gt;\n                &lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1581.svg\" alt=\"$\\displaystyle e^{iu} - e^{-iu} = 2i{\\mathrm {sen}}u. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"3P318\" title=\"3P318\"&gt;\n                Segue portanto que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\cos u = \\frac{e^{iu} + e^{-iu}}{2}, \\qquad \\text{e} \\qquad {\\mathrm {sen}}u = \\frac{e^{iu} - e^{-iu}}{2i}.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P319\" title=\"3P319\"&gt;\n                &lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1582.svg\" alt=\"$\\displaystyle \\cos u = \\frac{e^{iu} + e^{-iu}}{2},$\" loading=\"lazy\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1583.svg\" alt=\"$\\displaystyle \\qquad {\\mathrm {sen}}u = \\frac{e^{iu} - e^{-iu}}{2i}. $\" loading=\"lazy\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"3P320\" title=\"3P320\"&gt;\n                Estas duas igualdades são as identidades exponenciais para as funções trigonométricas circulares e são válidas para\n                valores reais de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1.svg\" alt=\"$u$\" loading=\"lazy\"&gt;&lt;/span&gt;. Obviamente combinando estas duas fórmulas, podemos deduzir fórmulas exponenciais para as outras\n                funções trigonométricas circulares. São elas\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"3P321\" title=\"3P321\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img1584.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.06ex; vertical-align: -1.72ex; \" src=\"img/img1585.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {sen}}u}{\\cos u} = \\frac{e^{iu} - e^{-iu}}{2i} \\cdot \\frac{2}{e^{iu} + e^{-iu}} = \\frac{ie^{-iu} - ie^{iu}}{e^{iu} + e^{-iu}},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.58ex; \" src=\"img/img1586.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.06ex; vertical-align: -1.72ex; \" src=\"img/img1587.svg\" alt=\"$\\displaystyle = \\frac{\\cos u}{{\\mathrm {sen}}u} = \\frac{e^{iu} + e^{-iu}}{2} \\cdot \\frac{2i}{e^{iu} - e^{-iu}} = \\frac{ie^{iu} + ie^{-iu}}{e^{iu} - e^{-iu}},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1588.svg\" alt=\"$\\displaystyle \\sec u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1589.svg\" alt=\"$\\displaystyle = \\frac{1}{\\cos u} = \\frac{2}{e^{iu} + e^{-iu}},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.19ex; vertical-align: -0.13ex; \" src=\"img/img1590.svg\" alt=\"$\\displaystyle \\csc u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1591.svg\" alt=\"$\\displaystyle = \\frac{1}{{\\mathrm {sen}}u} = \\frac{2i}{e^{iu} - e^{-iu}},$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                &lt;p class=\" unidade\" id=\"3P322\" style=\"text-indent: 0 !important;\" title=\"3P322\"&gt;\n                respeitados os domínios de definição das funções.\n                &lt;/p&gt;",
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      "Capítulo 3: Igualdades exponenciais e logarítmicas"
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    "title": "Capítulo 3: Igualdades exponenciais e logarítmicas",
    "section": "3.6 Fórmulas logarítmicas para as funções trigonométricas circulares inversas",
    "text": "3.6 Fórmulas logarítmicas para as funções trigonométricas circulares inversas\n\n&lt;p class=\" unidade\" id=\"3P323\" title=\"3P323\"&gt;\nPodemos também, como no caso hiperbólico, escrever as funções inversas das funções trigonométricas circulares em termos\ndo logaritmo. Isto porque a função logaritmo de um número complexo, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.21ex; vertical-align: -0.13ex; \" src=\"img/img1592.svg\" alt=\"$\\ln z$\" loading=\"lazy\"&gt;&lt;/span&gt; é a função inversa da exponencial &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.10ex; \" src=\"img/img1545.svg\" alt=\"$e^{z}$\" loading=\"lazy\"&gt;&lt;/span&gt;,\ncom uma certa restrição no logaritmo. Para conhecer mais sobre esta restrição, recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#ZillVC\"&gt;11&lt;/a&gt;, Zill].  Por\nhora, é suficiente saber que &lt;!-- MATH\n $\\ln e^{(a+ib)} = (a+ib)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1593.svg\" alt=\"$\\ln e^{(a+ib)} = (a+ib)$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n $a \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n $b \\in (-\\pi,\\pi]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1594.svg\" alt=\"$b \\in (-\\pi,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P324\" title=\"3P324\"&gt;\nTambém temos que lembrar que a conhecida “fórmula de Bháskara” continua válida para resolver equações quadráticas\nque envolvem coeficientes complexos. Mais precisamente, se &lt;!-- MATH\n $az^{2}+bz+c = 0$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img1595.svg\" alt=\"$az^{2}+bz+c = 0$\" loading=\"lazy\"&gt;&lt;/span&gt; é uma equação com &lt;!-- MATH\n $a, b, c \\in \\mathbb{C}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1596.svg\" alt=\"$a, b, c \\in \\mathbb{C}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img939.svg\" alt=\"$a \\neq 0$\" loading=\"lazy\"&gt;&lt;/span&gt;, então as soluções desta equação são dadas por\n&lt;!-- MATH\n \\begin{displaymath}\nz = \\frac{-b + \\sqrt{b^{2} - 4ac}}{2a}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P325\" title=\"3P325\"&gt;\n&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1597.svg\" alt=\"$\\displaystyle z = \\frac{-b + \\sqrt{b^{2} - 4ac}}{2a}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P326\" title=\"3P326\"&gt;\nNote que não usamos o sinal &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.60ex; vertical-align: -0.27ex; \" src=\"img/img1598.svg\" alt=\"$\\pm$\" loading=\"lazy\"&gt;&lt;/span&gt;, porque a função raiz quadrada (potência meio) para números complexos é bivalente,\nisto é, assume dois valores, que são simétricos com relação à origem e isto substitui o sinal &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.60ex; vertical-align: -0.27ex; \" src=\"img/img1598.svg\" alt=\"$\\pm$\" loading=\"lazy\"&gt;&lt;/span&gt;. Veja\n[&lt;a href=\"/trigonometria-hiperbolica/referencias#ZillVC\"&gt;11&lt;/a&gt;, Zill] ou outro texto sobre números complexos para um estudo mais completo sobre raízes de um número\ncomplexo.\n&lt;/p&gt;\n&lt;p class=\" unidade\" id=\"3P327\" title=\"3P327\"&gt;\nConsiderando &lt;!-- MATH\n $w = {\\mathrm {sen}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img458.svg\" alt=\"$w = {\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n $u \\in [-1,1]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n $w \\in [-\\frac{\\pi}{2},\\frac{\\pi}{2}]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img424.svg\" alt=\"$w \\in\n[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;, então temos a relação\n&lt;!-- MATH\n \\begin{displaymath}\nu = {\\mathrm {sen}}w = \\frac{e^{iw} - e^{-iw}}{2i},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P328\" title=\"3P328\"&gt;\n&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1599.svg\" alt=\"$\\displaystyle u = {\\mathrm {sen}}w = \\frac{e^{iw} - e^{-iw}}{2i}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P329\" style=\"text-indent: 0 !important;\" title=\"3P329\"&gt;\ne multiplicando esta igualdade por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1600.svg\" alt=\"$2i e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt; e, organizando os termos, temos\n&lt;!-- MATH\n \\begin{displaymath}\n(e^{iw})^{2} - 2iue^{iw} - 1 = 0.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P330\" title=\"3P330\"&gt;\n&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1601.svg\" alt=\"$\\displaystyle (e^{iw})^{2} - 2iue^{iw} - 1 = 0. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P331\" title=\"3P331\"&gt;\nResolvendo esta equação quadrática, em termos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt;, segue que\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\frac{2ui + \\sqrt{-4u^{2} + 4}}{2} = ui + \\sqrt{1-u^{2}}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P332\" title=\"3P332\"&gt;\n&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1603.svg\" alt=\"$\\displaystyle e^{iw} = \\frac{2ui + \\sqrt{-4u^{2} + 4}}{2} = ui + \\sqrt{1-u^{2}}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P333\" title=\"3P333\"&gt;\nComo &lt;!-- MATH\n $u \\in [-1,1]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; a raiz do segundo membro é um número real. Levando em conta que &lt;!-- MATH\n $w \\in\n[-\\frac{\\pi}{2},\\frac{\\pi}{2}]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img424.svg\" alt=\"$w \\in\n[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt; então aplicando o logaritmo em ambos os membros, obtemos\n&lt;!-- MATH\n \\begin{displaymath}\niw = \\ln e^{iw} =  \\ln(ui + \\sqrt{1-u^{2}}),\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P334\" title=\"3P334\"&gt;\n&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img1604.svg\" alt=\"$\\displaystyle iw = \\ln e^{iw} = \\ln(ui + \\sqrt{1-u^{2}}), $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P335\" style=\"text-indent: 0 !important;\" title=\"3P335\"&gt;\ne multiplicando tudo por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1605.svg\" alt=\"$-i$\" loading=\"lazy\"&gt;&lt;/span&gt;, segue que\n&lt;!-- MATH\n \\begin{displaymath}\n{\\mathrm {sen}}^{-1} u = w = -i \\ln(ui + \\sqrt{1-u^{2}}).\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P336\" title=\"3P336\"&gt;\n&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img1606.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}^{-1} u = w = -i \\ln(ui + \\sqrt{1-u^{2}}). $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P337\" title=\"3P337\"&gt;\nPara &lt;!-- MATH\n $w = \\cos^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img467.svg\" alt=\"$w = \\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, vale a relação &lt;!-- MATH\n $u = \\cos w = \\frac{e^{iw}+e^{-iw}}{2}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.09ex; vertical-align: -0.83ex; \" src=\"img/img1607.svg\" alt=\"$u = \\cos w = \\frac{e^{iw}+e^{-iw}}{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;, com &lt;!-- MATH\n $u \\in [-1,1]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n $w \\in [0,\\pi]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img564.svg\" alt=\"$w \\in [0,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;.\nProcedendo como anteriormente, multiplicamos esta relação por 2 e por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt;. Obtemos\n&lt;!-- MATH\n \\begin{displaymath}\n2ue^{iw} = (e^{iw})^{2} + 1,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P338\" title=\"3P338\"&gt;\n&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1608.svg\" alt=\"$\\displaystyle 2ue^{iw} = (e^{iw})^{2} + 1, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P339\" style=\"text-indent: 0 !important;\" title=\"3P339\"&gt;\ne resolvendo a equação quadrática\n&lt;!-- MATH\n \\begin{displaymath}\n(e^{iw})^{2} - 2ue^{iw} + 1 = 0\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P340\" title=\"3P340\"&gt;\n&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1609.svg\" alt=\"$\\displaystyle (e^{iw})^{2} - 2ue^{iw} + 1 = 0 $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P341\" style=\"text-indent: 0 !important;\" title=\"3P341\"&gt;\nem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\frac{2u + \\sqrt{4u^{2} - 4}}{2} = u + \\sqrt{u^{2} - 1}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P342\" title=\"3P342\"&gt;\n&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1610.svg\" alt=\"$\\displaystyle e^{iw} = \\frac{2u + \\sqrt{4u^{2} - 4}}{2} = u + \\sqrt{u^{2} - 1}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P343\" title=\"3P343\"&gt;\nEsta solução envolve a raiz quadrada de um número que é real e negativo, já que &lt;!-- MATH\n $u \\in [-1,1]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img423.svg\" alt=\"$u \\in [-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Escrevemos então\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = u + i\\sqrt{1-u^{2}},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P344\" title=\"3P344\"&gt;\n&lt;img style=\"height: 2.80ex; vertical-align: -0.41ex; \" src=\"img/img1611.svg\" alt=\"$\\displaystyle e^{iw} = u + i\\sqrt{1-u^{2}}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P345\" style=\"text-indent: 0 !important;\" title=\"3P345\"&gt;\ne agora aplicando o logaritmo em ambos os membros, vem\n&lt;!-- MATH\n \\begin{displaymath}\niw = \\ln (u + i\\sqrt{1-u^{2}}),\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P346\" title=\"3P346\"&gt;\n&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img1612.svg\" alt=\"$\\displaystyle iw = \\ln (u + i\\sqrt{1-u^{2}}), $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P347\" style=\"text-indent: 0 !important;\" title=\"3P347\"&gt;\njá que &lt;!-- MATH\n $w \\in [0,\\pi]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img564.svg\" alt=\"$w \\in [0,\\pi]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Multiplicando a igualdade por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1605.svg\" alt=\"$-i$\" loading=\"lazy\"&gt;&lt;/span&gt; obtemos\n&lt;!-- MATH\n \\begin{displaymath}\n\\cos^{-1} u = w = -i \\ln (u + i\\sqrt{1-u^{2}}).\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P348\" title=\"3P348\"&gt;\n&lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img1613.svg\" alt=\"$\\displaystyle \\cos^{-1} u = w = -i \\ln (u + i\\sqrt{1-u^{2}}). $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P349\" title=\"3P349\"&gt;\nConsiderando agora &lt;!-- MATH\n $w = {\\mathrm  {tg}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img1614.svg\" alt=\"$w = {\\mathrm {tg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, válida para todo &lt;!-- MATH\n $u \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n $w \\in (-\\pi,\\pi)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1615.svg\" alt=\"$w \\in (-\\pi,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n&lt;!-- MATH\n \\begin{displaymath}\nu = {\\mathrm  {tg}}w = \\frac{ie^{-iw} - ie^{iw}}{e^{iw} + e^{-iw}}\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P350\" title=\"3P350\"&gt;\n&lt;img style=\"height: 5.06ex; vertical-align: -1.72ex; \" src=\"img/img1616.svg\" alt=\"$\\displaystyle u = {\\mathrm {tg}}w = \\frac{ie^{-iw} - ie^{iw}}{e^{iw} + e^{-iw}} $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P351\" style=\"text-indent: 0 !important;\" title=\"3P351\"&gt;\ndonde\n&lt;!-- MATH\n \\begin{displaymath}\nue^{iw} + ue^{-iw} = ie^{-iw} - ie^{iw}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P352\" title=\"3P352\"&gt;\n&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1617.svg\" alt=\"$\\displaystyle ue^{iw} + ue^{-iw} = ie^{-iw} - ie^{iw}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P353\" title=\"3P353\"&gt;\nMultiplicando por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt; e organizando os termos temos a equação quadrática\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P354\" title=\"3P354\"&gt;&lt;table class=\"equation\"&gt;\n&lt;tbody&gt;&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1618.svg\" alt=\"$\\displaystyle u(e^{iw})^{2} + u = i - i(e^{iw})^{2}$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;tr&gt;\n&lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1619.svg\" alt=\"$\\displaystyle (i+u)(e^{iw})^{2} - (i-u) = 0.$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n&nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n\n\n&lt;p class=\" unidade\" id=\"3P355\" title=\"3P355\"&gt;\nResolvendo em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt; obtemos,\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\sqrt{\\tfrac{i-u}{i+u}} = \\left( \\tfrac{i-u}{i+u} \\right)^{\\frac{1}{2}},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P356\" title=\"3P356\"&gt;\n&lt;img style=\"height: 4.08ex; vertical-align: -1.28ex; \" src=\"img/img1620.svg\" alt=\"$\\displaystyle e^{iw} = \\sqrt{\\tfrac{i-u}{i+u}} = \\left( \\tfrac{i-u}{i+u} \\right)^{\\frac{1}{2}}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P357\" style=\"text-indent: 0 !important;\" title=\"3P357\"&gt;\ne aplicando logaritmo em ambos os membros, já que &lt;!-- MATH\n $w \\in (-\\pi,\\pi)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1615.svg\" alt=\"$w \\in (-\\pi,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n&lt;!-- MATH\n \\begin{displaymath}\niw = \\ln (e^{iw}) = \\ln\\left( \\tfrac{i-u}{i+u} \\right)^{\\frac{1}{2}} = \\frac{1}{2} \\ln \\left( \\tfrac{i-u}{i+u} \\right),\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P358\" title=\"3P358\"&gt;\n&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1621.svg\" alt=\"$\\displaystyle iw = \\ln (e^{iw}) = \\ln\\left( \\tfrac{i-u}{i+u} \\right)^{\\frac{1}{2}} = \\frac{1}{2} \\ln \\left( \\tfrac{i-u}{i+u} \\right), $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P359\" style=\"text-indent: 0 !important;\" title=\"3P359\"&gt;\ne multiplicando a igualdade por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1605.svg\" alt=\"$-i$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n&lt;!-- MATH\n \\begin{displaymath}\nw = \\frac{-i}{2} \\ln \\left( \\tfrac{i-u}{i+u} \\right) = \\frac{i}{2} \\ln \\left( \\tfrac{i+u}{i-u} \\right).\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P360\" title=\"3P360\"&gt;\n&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1622.svg\" alt=\"$\\displaystyle w = \\frac{-i}{2} \\ln \\left( \\tfrac{i-u}{i+u} \\right) = \\frac{i}{2} \\ln \\left( \\tfrac{i+u}{i-u} \\right). $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P361\" title=\"3P361\"&gt;\nAnalogamente para a cotangente inversa, temos &lt;!-- MATH\n $w = {\\mathrm {ctg}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img1623.svg\" alt=\"$w = {\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n $w \\in (0,\\pi)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img565.svg\" alt=\"$w \\in (0,\\pi)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n $u \\in \\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img57.svg\" alt=\"$u \\in \\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; e vale a relação\n&lt;!-- MATH\n \\begin{displaymath}\nu = {\\mathrm {ctg}}w = \\frac{ie^{iw} + ie^{-iw}}{e^{iw} - e^{-iw}}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P362\" title=\"3P362\"&gt;\n&lt;img style=\"height: 5.06ex; vertical-align: -1.72ex; \" src=\"img/img1624.svg\" alt=\"$\\displaystyle u = {\\mathrm {ctg}}w = \\frac{ie^{iw} + ie^{-iw}}{e^{iw} - e^{-iw}}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P363\" title=\"3P363\"&gt;\nProcedendo como no caso da tangente, chegamos a equação quadrática\n&lt;!-- MATH\n \\begin{displaymath}\n(u-i)(e^{iw})^{2} - (u+i) = 0,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P364\" title=\"3P364\"&gt;\n&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1625.svg\" alt=\"$\\displaystyle (u-i)(e^{iw})^{2} - (u+i) = 0, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P365\" style=\"text-indent: 0 !important;\" title=\"3P365\"&gt;\nque resolvida nos fornece,\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\sqrt{\\tfrac{u+i}{u-i}} = \\left( \\tfrac{u+i}{u-i} \\right)^{\\frac{1}{2}},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P366\" title=\"3P366\"&gt;\n&lt;img style=\"height: 4.08ex; vertical-align: -1.28ex; \" src=\"img/img1626.svg\" alt=\"$\\displaystyle e^{iw} = \\sqrt{\\tfrac{u+i}{u-i}} = \\left( \\tfrac{u+i}{u-i} \\right)^{\\frac{1}{2}}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P367\" title=\"3P367\"&gt;\nAplicando o logaritmo e multiplicando o resultado por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1605.svg\" alt=\"$-i$\" loading=\"lazy\"&gt;&lt;/span&gt;, chegamos a\n&lt;!-- MATH\n \\begin{displaymath}\n{\\mathrm {ctg}}^{-1} u = w = -i \\ln \\left( \\tfrac{u+i}{u-i} \\right)^{\\frac{1}{2}} = \\frac{-i}{2} \\ln \\left( \\tfrac{u+i}{u-i} \\right) = \\frac{i}{2} \\ln \\left( \\tfrac{u-i}{u+i} \\right) .\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P368\" title=\"3P368\"&gt;\n&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1627.svg\" alt=\"$\\displaystyle {\\mathrm {ctg}}^{-1} u = w = -i \\ln \\left( \\tfrac{u+i}{u-i} \\righ...\n...( \\tfrac{u+i}{u-i} \\right) = \\frac{i}{2} \\ln \\left( \\tfrac{u-i}{u+i} \\right) . $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P369\" title=\"3P369\"&gt;\nConsiderando agora &lt;!-- MATH\n $w = \\sec^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img1628.svg\" alt=\"$w = \\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, para todo &lt;!-- MATH\n $u \\in [1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1075.svg\" alt=\"$u \\in [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; e &lt;!-- MATH\n $w \\in [0,\\frac{\\pi}{2})$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1629.svg\" alt=\"$w \\in [0,\\frac{\\pi}{2})$\" loading=\"lazy\"&gt;&lt;/span&gt;. Tomamos a relação &lt;!-- MATH\n $u =\n\\sec w = \\frac{2}{e^{iw} + e^{-iw}}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img1630.svg\" alt=\"$u =\n\\sec w = \\frac{2}{e^{iw} + e^{-iw}}$\" loading=\"lazy\"&gt;&lt;/span&gt; e obtemos,\n&lt;!-- MATH\n \\begin{displaymath}\nue^{iw} + ue^{-iw} = 2.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P370\" title=\"3P370\"&gt;\n&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1631.svg\" alt=\"$\\displaystyle ue^{iw} + ue^{-iw} = 2. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P371\" title=\"3P371\"&gt;\nMultiplicando a equação por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt; e reorganizando os termos obtemos\n&lt;!-- MATH\n \\begin{displaymath}\nu(e^{iw})^{2} - 2e^{iw} + u = 0,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P372\" title=\"3P372\"&gt;\n&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1632.svg\" alt=\"$\\displaystyle u(e^{iw})^{2} - 2e^{iw} + u = 0, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P373\" style=\"text-indent: 0 !important;\" title=\"3P373\"&gt;\nque resolvida em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt; nos leva a\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\frac{2 + \\sqrt{4 - 4u^{2}}}{2u} = \\frac{1+\\sqrt{1-u^{2}}}{u}.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P374\" title=\"3P374\"&gt;\n&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1633.svg\" alt=\"$\\displaystyle e^{iw} = \\frac{2 + \\sqrt{4 - 4u^{2}}}{2u} = \\frac{1+\\sqrt{1-u^{2}}}{u}. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P375\" title=\"3P375\"&gt;\nA raiz quadrada do segundo membro tem no radicando um número real negativo, já que &lt;!-- MATH\n $u \\in [1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1075.svg\" alt=\"$u \\in [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt;. Escrevemos então\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\frac{1+i\\sqrt{u^{2}-1}}{u},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P376\" title=\"3P376\"&gt;\n&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1634.svg\" alt=\"$\\displaystyle e^{iw} = \\frac{1+i\\sqrt{u^{2}-1}}{u}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P377\" style=\"text-indent: 0 !important;\" title=\"3P377\"&gt;\ne temos\n&lt;!-- MATH\n \\begin{displaymath}\niw = \\ln(e^{iw}) = \\ln \\left( \\tfrac{1+i\\sqrt{u^{2}-1}}{u} \\right),\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P378\" title=\"3P378\"&gt;\n&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1635.svg\" alt=\"$\\displaystyle iw = \\ln(e^{iw}) = \\ln \\left( \\tfrac{1+i\\sqrt{u^{2}-1}}{u} \\right), $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P379\" style=\"text-indent: 0 !important;\" title=\"3P379\"&gt;\nou ainda,\n&lt;!-- MATH\n \\begin{displaymath}\nw = -i \\ln \\left( \\tfrac{1+i\\sqrt{u^{2}-1}}{u} \\right).\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P380\" title=\"3P380\"&gt;\n&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1636.svg\" alt=\"$\\displaystyle w = -i \\ln \\left( \\tfrac{1+i\\sqrt{u^{2}-1}}{u} \\right). $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P381\" title=\"3P381\"&gt;\nFinalmente, para &lt;!-- MATH\n $w = \\csc^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img1637.svg\" alt=\"$w = \\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;, fazendo &lt;!-- MATH\n $u = \\csc w = \\frac{2i}{e^{iw} - e^{-iw}}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img1638.svg\" alt=\"$u = \\csc w = \\frac{2i}{e^{iw} - e^{-iw}}$\" loading=\"lazy\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n $u \\in [1, \\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1075.svg\" alt=\"$u \\in [1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; e\n&lt;!-- MATH\n $w \\in (0, \\frac{\\pi}{2}]$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1639.svg\" alt=\"$w \\in (0, \\frac{\\pi}{2}]$\" loading=\"lazy\"&gt;&lt;/span&gt;. Temos então\n&lt;!-- MATH\n \\begin{displaymath}\nue^{iw} - ue^{-iw} = 2i,\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P382\" title=\"3P382\"&gt;\n&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1640.svg\" alt=\"$\\displaystyle ue^{iw} - ue^{-iw} = 2i, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P383\" style=\"text-indent: 0 !important;\" title=\"3P383\"&gt;\ne então\n&lt;!-- MATH\n \\begin{displaymath}\nu(e^{iw})^{2} -2ie^{iw} - u = 0.\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P384\" title=\"3P384\"&gt;\n&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1641.svg\" alt=\"$\\displaystyle u(e^{iw})^{2} -2ie^{iw} - u = 0. $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P385\" title=\"3P385\"&gt;\nResolvendo em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1602.svg\" alt=\"$e^{iw}$\" loading=\"lazy\"&gt;&lt;/span&gt;, vem\n&lt;!-- MATH\n \\begin{displaymath}\ne^{iw} = \\frac{2i + \\sqrt{-4 + 4u^{2}}}{2u} = \\frac{i + \\sqrt{u^{2}-1}}{u},\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P386\" title=\"3P386\"&gt;\n&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img1642.svg\" alt=\"$\\displaystyle e^{iw} = \\frac{2i + \\sqrt{-4 + 4u^{2}}}{2u} = \\frac{i + \\sqrt{u^{2}-1}}{u}, $\" loading=\"lazy\"&gt;\n&lt;/div&gt;&lt;p class=\" unidade\" id=\"3P387\" style=\"text-indent: 0 !important;\" title=\"3P387\"&gt;\ne aplicando o logaritmo e uma multiplicação por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1605.svg\" alt=\"$-i$\" loading=\"lazy\"&gt;&lt;/span&gt;, temos\n&lt;!-- MATH\n \\begin{displaymath}\n\\csc^{-1} u = w = -i \\ln \\left(\\tfrac{i + \\sqrt{u^{2}-1}}{u} \\right).\n\\end{displaymath}\n --&gt;\n&lt;/p&gt;\n&lt;div class=\"mathdisplay unidade\" id=\"3P388\" title=\"3P388\"&gt;\n&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1643.svg\" alt=\"$\\displaystyle \\csc^{-1} u = w = -i \\ln \\left(\\tfrac{i + \\sqrt{u^{2}-1}}{u} \\right). $\" loading=\"lazy\"&gt;\n&lt;/div&gt;\n\n&lt;p class=\" unidade\" id=\"3P389\" title=\"3P389\"&gt;\nOrganizando as fórmulas logarítmicas, temos a tabela abaixo.\n&lt;/p&gt;\n\n&lt;div class=\"CENTER\"&gt;&lt;a id=\"5383\"&gt;&lt;/a&gt;\n&lt;table&gt;\n&lt;caption&gt;&lt;strong&gt;Tabela 3.2:&lt;/strong&gt;\nFórmulas logarítmicas para as funções trigonométricas circulares inversas.&lt;/caption&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n&lt;div class=\"CENTER\"&gt;\n&lt;table class=\"PAD  BORDER\"&gt;\n&lt;tbody&gt;&lt;tr&gt;&lt;td class=\"CENTER\"&gt;função&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;domínio&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;fórmula logarítmica&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n ${\\mathrm {sen}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img428.svg\" alt=\"${\\mathrm {sen}}^{-1}u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $-i \\ln(ui + \\sqrt{1-u^{2}})$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img1644.svg\" alt=\"$-i \\ln(ui + \\sqrt{1-u^{2}})$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n&lt;br&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\cos^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img521.svg\" alt=\"$\\cos^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $-i \\ln (u + i\\sqrt{1-u^{2}})$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img1645.svg\" alt=\"$-i \\ln (u + i\\sqrt{1-u^{2}})$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n&lt;br&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n ${\\mathrm  {tg}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img522.svg\" alt=\"${\\mathrm {tg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\frac{i}{2} \\ln \\left( \\tfrac{i+u}{i-u} \\right)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img1646.svg\" alt=\"$\\frac{i}{2} \\ln \\left( \\tfrac{i+u}{i-u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n&lt;br&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n ${\\mathrm {ctg}}^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img523.svg\" alt=\"${\\mathrm {ctg}}^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n $\\mathbb{R}$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\frac{i}{2} \\ln \\left( \\tfrac{u-i}{u+i} \\right)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.94ex; vertical-align: -0.95ex; \" src=\"img/img1647.svg\" alt=\"$\\frac{i}{2} \\ln \\left( \\tfrac{u-i}{u+i} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n&lt;br&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\sec^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img524.svg\" alt=\"$\\sec^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n $[1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $-i \\ln \\left( \\tfrac{1+i\\sqrt{u^{2}-1}}{u} \\right)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1648.svg\" alt=\"$-i \\ln \\left( \\tfrac{1+i\\sqrt{u^{2}-1}}{u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt; &lt;br&gt;\n&lt;br&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $\\csc^{-1} u$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.13ex; \" src=\"img/img525.svg\" alt=\"$\\csc^{-1} u$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;!-- MATH\n $[1,\\infty)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img547.svg\" alt=\"$[1,\\infty)$\" loading=\"lazy\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n&lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n $-i \\ln \\left(\\tfrac{i + \\sqrt{u^{2}-1}}{u} \\right)$\n --&gt;\n&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1649.svg\" alt=\"$-i \\ln \\left(\\tfrac{i + \\sqrt{u^{2}-1}}{u} \\right)$\" loading=\"lazy\"&gt;&lt;/span&gt;&lt;/td&gt;\n&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n&lt;/tbody&gt;&lt;/table&gt;\n&lt;/div&gt;",
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      "Capítulo 3: Igualdades exponenciais e logarítmicas"
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    "title": "Capítulo 4: Aplicações",
    "section": "4.1 Cálculo do número \\(\\pi\\)",
    "text": "4.1 Cálculo do número \\(\\pi\\)\n\n        &lt;p class=\" unidade\" id=\"4P4\" title=\"4P4\"&gt;\n        O número “pi”, é conhecido da humanidade ainda antes de Cristo. É difícil dizer com precisão quando foi concebido,\n        mas desde muito cedo, o homem percebeu que dividindo o comprimento de uma circunferência qualquer pelo seu diâmetro,\n        resultava sempre um mesmo valor.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P5\" title=\"4P5\"&gt;\n        O símbolo atual que designa o número “pi” é a letra grega &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;, que foi utilizada pela primeira vez em 1706 por\n        William Jones, mas só foi amplamente aceita quando usada por Euler em 1737. Fato este que não nos impedirá de usar a\n        notação atual &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;, mesmo para citações mais antigas.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P6\" title=\"4P6\"&gt;\n        O primeiro matemático a investigar o número &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; foi Arquimedes (287-212 a.C.). Ele efetivamente calculou uma\n        aproximação para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;. Arquimedes construiu polígonos regulares inscritos e circunscritos em uma circunferência e\n        calculou o perímetro destes polígonos. Quanto mais lados ele colocava no polígono, melhor a aproximação. Usando um\n        polígono regular de 96 lados, Arquimedes afirmou que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        3 + \\frac{10}{71} &lt; \\pi &lt; 3 + \\frac{10}{70},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P7\" title=\"4P7\"&gt;\n        &lt;img style=\"height: 4.55ex; vertical-align: -1.57ex; \" src=\"img/img1650.svg\" alt=\"$\\displaystyle 3 + \\frac{10}{71} &lt; \\pi &lt; 3 + \\frac{10}{70}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P8\" style=\"text-indent: 0 !important;\" title=\"4P8\"&gt;\n        ou seja, &lt;!-- MATH\n         $\\frac{223}{71} &lt; \\pi &lt; \\frac{22}{7}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.82ex; vertical-align: -0.83ex; \" src=\"img/img1651.svg\" alt=\"$\\frac{223}{71} &lt; \\pi &lt; \\frac{22}{7}$\"&gt;&lt;/span&gt;. A fração &lt;!-- MATH\n         $\\frac{22}{7}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1652.svg\" alt=\"$\\frac{22}{7}$\"&gt;&lt;/span&gt; é uma das mais famosas aproximações para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;.\n        Entretanto, um artifício do Cálculo Diferencial e Integral nos mostra que &lt;!-- MATH\n         $\\pi \\neq \\frac{22}{7}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1653.svg\" alt=\"$\\pi \\neq \\frac{22}{7}$\"&gt;&lt;/span&gt;. Mais precisamente,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        0 &lt; \\int_{0}^{1} \\frac{x^4(1-x)^4}{1+x^2} dx\n        = \\frac{22}{7}-\\pi. \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P9\" title=\"4P9\"&gt;\n        &lt;img style=\"height: 5.44ex; vertical-align: -2.03ex; \" src=\"img/img1654.svg\" alt=\"$\\displaystyle 0 &lt; \\int_{0}^{1} \\frac{x^4(1-x)^4}{1+x^2} dx\n        = \\frac{22}{7}-\\pi. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P10\" title=\"4P10\"&gt;\n        Arquimedes, assim como outros matemáticos de sua época, acreditava que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; fosse um número racional. No entanto, em 1761 o\n        alemão Johann Lambert provou que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; é um número irracional. Isto significa que esse número, assim como todos os\n        números irracionais, possui infinitas casas decimais que não apresentam comportamento periódico.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P11\" title=\"4P11\"&gt;\n        Devido a este fato, vários matemáticos ficaram ocupados durante algum tempo para calcular o valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; com mais casas decimais corretas. O objetivo desta seção é mostrar como a função arco tangente pode ser utilizada para calcular\n        casas decimais do número &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P12\" title=\"4P12\"&gt;\n        Para iniciarmos, precisamos fazer algumas apresentações a respeito de séries geométricas e séries alternadas. É\n        recomendado ao leitor alguma habilidade sobre sequências e séries. Para um estudo mais aprofundado sobre estas e outras\n        séries recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#Swokowski\"&gt;8&lt;/a&gt;, Swokowski].\n        \n        &lt;/p&gt;\n        &lt;div&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;&lt;/b&gt; &nbsp; \n        Dada uma sequência geométrica infinita &lt;!-- MATH\n         $\\{ ar^{n} \\}_{n \\geq 0}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.37ex; vertical-align: -0.70ex; \" src=\"img/img1655.svg\" alt=\"$\\{ ar^{n} \\}_{n \\geq 0}$\"&gt;&lt;/span&gt;, a soma dos termos desta sequência é chamada de série\n        geométrica. É uma expressão da forma\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\sum_{n=0}^{\\infty} a r^{n} = a + a r + a r^{2} + a r^{3} + \\dots + a r^{n} + \\cdots.\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"4P13\" title=\"4P13\"&gt;\n        &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1656.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} a r^{n} = a + a r + a r^{2} + a r^{3} + \\dots + a r^{n} + \\cdots. $\"&gt;\n        &lt;/div&gt;&lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P14\" title=\"4P14\"&gt;\n        No caso de a soma infinita existir e ser igual a um número real &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img7.svg\" alt=\"$S$\"&gt;&lt;/span&gt;, então dizemos que a série é convergente, ou\n        ainda, convergente para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img7.svg\" alt=\"$S$\"&gt;&lt;/span&gt;. No caso em que a soma não existir então a série é dita divergente. Como casos\n        particulares, observe que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1376.svg\" alt=\"$r = 0$\"&gt;&lt;/span&gt; então a soma é igual a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\"&gt;&lt;/span&gt; e portanto convergente e se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1387.svg\" alt=\"$r = 1$\"&gt;&lt;/span&gt; então a soma é\n        &lt;!-- MATH\n         $\\infty \\cdot a = \\infty$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1657.svg\" alt=\"$\\infty \\cdot a = \\infty$\"&gt;&lt;/span&gt; e portanto divergente.\n        &lt;/p&gt;\n        \n        &lt;div id=\"4Teo2\" title=\"4Teo2\" class=\"unidade\"&gt;&lt;a id=\"teosgeo\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n        &lt;i&gt;Uma série geométrica, &lt;!-- MATH\n         $\\sum ar^{n}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1658.svg\" alt=\"$\\sum ar^{n}$\"&gt;&lt;/span&gt;, é convergente, se e somente se, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1659.svg\" alt=\"$\\vert r\\vert &lt; 1$\"&gt;&lt;/span&gt;. No caso de convergir, o valor desta\n        soma é precisamente o número &lt;!-- MATH\n         $S = \\frac{a}{1-r}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.58ex; vertical-align: -0.95ex; \" src=\"img/img1660.svg\" alt=\"$S = \\frac{a}{1-r}$\"&gt;&lt;/span&gt;.\n        &lt;/i&gt;&lt;/div&gt;\n        \n        &lt;div&gt;&lt;b&gt;Definição  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;&lt;/b&gt; &nbsp; \n        Se &lt;!-- MATH\n         $\\{ a_{n} \\}_{n \\geq 0}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.37ex; vertical-align: -0.70ex; \" src=\"img/img1661.svg\" alt=\"$\\{ a_{n} \\}_{n \\geq 0}$\"&gt;&lt;/span&gt; é uma sequência infinita de termos positivos, então uma série alternada é uma soma da\n        forma,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\sum_{n=0}^{\\infty} (-1)^{n} a_{n} = a_{0} - a_{1} + a_{2} - a_{3} + \\cdots + (-1)^{n} a_{n} + \\cdots\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"4P15\" title=\"4P15\"&gt;\n        &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1662.svg\" alt=\"$\\displaystyle \\sum_{n=0}^{\\infty} (-1)^{n} a_{n} = a_{0} - a_{1} + a_{2} - a_{3} + \\cdots + (-1)^{n} a_{n} + \\cdots $\"&gt;\n        &lt;/div&gt;&lt;/div&gt;\n        \n        \n        &lt;div id=\"4Teo4\" title=\"4Teo4\" class=\"unidade\"&gt;&lt;a id=\"seriealt\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n        &lt;i&gt;Uma série alternada &lt;!-- MATH\n         $\\sum (-1)^{n}a_{n}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1663.svg\" alt=\"$\\sum (-1)^{n}a_{n}$\"&gt;&lt;/span&gt; é convergente se os termos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\"&gt;&lt;/span&gt; formam uma sequência positiva, decrescente,\n        e que tende a zero.\n        &lt;/i&gt;&lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P16\" title=\"4P16\"&gt;\n        O leitor interessado nas demonstrações dos dois últimos teoremas, ou em alguns exemplos de séries alternadas e\n        geométricas, pode consultar [&lt;a href=\"/trigonometria-hiperbolica/referencias#Swokowski\"&gt;8&lt;/a&gt;, Swokowski].\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P17\" title=\"4P17\"&gt;\n        Consideremos então que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; seja uma variável real que assume valores no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\"&gt;&lt;/span&gt;. Então &lt;!-- MATH\n         $x^{2} \\in [0,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1664.svg\" alt=\"$x^{2} \\in [0,1)$\"&gt;&lt;/span&gt;.\n        Podemos assim, construir uma série geométrica com primeiro termo igual a 1 e razão &lt;!-- MATH\n         $r = -x^{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img1665.svg\" alt=\"$r = -x^{2}$\"&gt;&lt;/span&gt;, que de acordo com o teorema &lt;a href=\"#teosgeo\"&gt;4.2&lt;/a&gt;, é convergente para o número &lt;!-- MATH\n         $S = \\frac{1}{1-r} = \\frac{1}{1+x^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img1666.svg\" alt=\"$S = \\frac{1}{1-r} = \\frac{1}{1+x^{2}}$\"&gt;&lt;/span&gt;, já que &lt;!-- MATH\n         $|r| = x^{2} &lt; 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1667.svg\" alt=\"$\\vert r\\vert = x^{2} &lt; 1$\"&gt;&lt;/span&gt;.\n        Temos então que,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P18\" title=\"4P18\"&gt;&lt;a id=\"seriepot\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        \\frac{1}{1+x^{2}} = \\sum_{n=0}^{\\infty} (-1)^{n}(x^{2})^{n} = 1 - x^{2} + x^{4} - x^{6} + \\cdots + (-1)^{n}x^{2n} + \\cdots,\n        \\end{equation}\n         --&gt;\n        &lt;table&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1668.svg\" alt=\"$\\displaystyle \\frac{1}{1+x^{2}} = \\sum_{n=0}^{\\infty} (-1)^{n}(x^{2})^{n} = 1 - x^{2} + x^{4} - x^{6} + \\cdots + (-1)^{n}x^{2n} + \\cdots,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;1&lt;/span&gt;)\n        &lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        \n    &lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"4P19\" style=\"text-indent: 0 !important;\" title=\"4P19\"&gt;\n        para qualquer &lt;!-- MATH\n         $x \\in (-1,1)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1277.svg\" alt=\"$x \\in (-1,1)$\"&gt;&lt;/span&gt;. Observe que esta é uma série geométrica, mas também é uma série alternada. Podemos ainda\n        dizer que esta série é uma série de potências de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, uma vez que seus termos são potências da variável &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P20\" title=\"4P20\"&gt;\n        Lembremos agora que a fração &lt;!-- MATH\n         $\\frac{1}{1+x^{2}}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.05ex; vertical-align: -1.06ex; \" src=\"img/img1669.svg\" alt=\"$\\frac{1}{1+x^{2}}$\"&gt;&lt;/span&gt;, vista como função de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, é a derivada da função &lt;!-- MATH\n         $y = {\\mathrm  {tg}}^{-1}(x)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1670.svg\" alt=\"$y = {\\mathrm {tg}}^{-1}(x)$\"&gt;&lt;/span&gt;,\n        exatamente como vimos na seção (&lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#secdercircinv\"&gt;1.7&lt;/a&gt;). Em outras palavras,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{d}{dx} ({\\mathrm  {tg}}^{-1} x) = \\frac{1}{1+x^{2}},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P21\" title=\"4P21\"&gt;\n        &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1671.svg\" alt=\"$\\displaystyle \\frac{d}{dx} ({\\mathrm {tg}}^{-1} x) = \\frac{1}{1+x^{2}}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P22\" style=\"text-indent: 0 !important;\" title=\"4P22\"&gt;\n        e isto significa que,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}^{-1} x = \\int \\frac{1}{1+x^{2}} dx = \\int \\left( \\sum_{n=0}^{\\infty} (-1)^{n} x^{2n} \\right) dx.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P23\" title=\"4P23\"&gt;\n        &lt;img style=\"height: 6.68ex; vertical-align: -2.91ex; \" src=\"img/img1672.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1} x = \\int \\frac{1}{1+x^{2}} dx = \\int \\left( \\sum_{n=0}^{\\infty} (-1)^{n} x^{2n} \\right) dx. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P24\" title=\"4P24\"&gt;\n        Recorremos agora ao teorema que garante a integração de uma série de potências. A demonstração deste resultado também\n        pode ser encontrada em [&lt;a href=\"/trigonometria-hiperbolica/referencias#Swokowski\"&gt;8&lt;/a&gt;, Swokowski].\n        &lt;/p&gt;\n        \n        &lt;div id=\"4Teo5\" title=\"4Teo5\" class=\"unidade\"&gt;&lt;a id=\"teointserie\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n        &lt;i&gt;Se uma função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\"&gt;&lt;/span&gt; possui representação em série de potência de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, isto é, &lt;!-- MATH\n         $f(x) = \\sum a_{n} x^{n}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1132.svg\" alt=\"$f(x) = \\sum a_{n}x^{n}$\"&gt;&lt;/span&gt; e esta série\n        for convergente em todo &lt;!-- MATH\n         $x \\in (-c,c) \\subset \\mathbb{R}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1673.svg\" alt=\"$x \\in (-c,c) \\subset \\mathbb{R}$\"&gt;&lt;/span&gt;, então\n        &lt;/i&gt;&lt;!-- MATH\n         \\begin{displaymath}\n        \\int f(x) dx = \\int \\left( \\sum_{n=0}^{\\infty} a_{n} x^{n} \\right) dx = \\sum_{n=0}^{\\infty} \\left( \\int a_{n} x^{n} dx \\right),\n        \\end{displaymath}\n         --&gt;\n        \n        &lt;div class=\"mathdisplay unidade\" id=\"4P25\" title=\"4P25\"&gt;\n        &lt;img style=\"height: 6.68ex; vertical-align: -2.91ex; \" src=\"img/img1674.svg\" alt=\"$\\displaystyle \\int f(x) dx = \\int \\left( \\sum_{n=0}^{\\infty} a_{n} x^{n} \\right) dx = \\sum_{n=0}^{\\infty} \\left( \\int a_{n} x^{n} dx \\right), $\"&gt;\n        &lt;/div&gt;&lt;i&gt;\n        para todo &lt;!-- MATH\n         $x \\in (-c,c)$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1675.svg\" alt=\"$x \\in (-c,c)$\"&gt;&lt;/span&gt;. A convergência da nova série obtida pela integração dos termos, pode ser alterada se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.60ex; vertical-align: -0.27ex; \" src=\"img/img1676.svg\" alt=\"$x = \\pm\n        c$\"&gt;&lt;/span&gt;.\n        &lt;/i&gt;&lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P26\" title=\"4P26\"&gt;\n        Este teorema nos permite então determinar, por integração, a série da função arco tangente. Temos assim,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P27\" title=\"4P27\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.61ex; vertical-align: -0.58ex; \" src=\"img/img1677.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1} x$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.68ex; vertical-align: -2.91ex; \" src=\"img/img1678.svg\" alt=\"$\\displaystyle = \\int \\frac{1}{1+x^{2}} dx = \\int \\left( \\sum_{n=0}^{\\infty} (-1)^{n} x^{2n} \\right) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1679.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\left( \\int (-1)^{n} x^{2n} dx \\right)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1680.svg\" alt=\"$\\displaystyle = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n} x^{2n+1}}{2n+1}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.06ex; vertical-align: -1.72ex; \" src=\"img/img1681.svg\" alt=\"$\\displaystyle = x - \\frac{x^{3}}{3} + \\frac{x^{5}}{5} - \\frac{x^{7}}{7} + \\cdots + \\frac{(-1)^{n} x^{2n+1}}{2n+1} + \\cdots$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"4P28\" style=\"text-indent: 0 !important;\" title=\"4P28\"&gt;\n        e, assim, obtemos a igualdade desejada para o nosso objetivo,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P29\" title=\"4P29\"&gt;&lt;a id=\"arctanserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        {\\mathrm  {tg}}^{-1} x = x - \\frac{x^{3}}{3} + \\frac{x^{5}}{5} - \\frac{x^{7}}{7} + \\frac{x^{9}}{9} - \\frac{x^{11}}{11} + \\cdots +\n        \\frac{(-1)^{n} x^{2n+1}}{2n+1} + \\cdots.\n        \\end{equation}\n         --&gt;\n        &lt;table&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.06ex; vertical-align: -1.72ex; \" src=\"img/img1682.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1} x = x - \\frac{x^{3}}{3} + \\frac{x^{5}}{5} - \\...\n        ...{9}}{9} - \\frac{x^{11}}{11} + \\cdots +\n        \\frac{(-1)^{n} x^{2n+1}}{2n+1} + \\cdots.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;2&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n       \n    &lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P30\" title=\"4P30\"&gt;\n        Esta igualdade é válida para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img614.svg\" alt=\"$(-1,1)$\"&gt;&lt;/span&gt;. O ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\"&gt;&lt;/span&gt;, deve ser avaliado novamente pois é um dos\n        extremos do intervalo de convergência. Para ser mais preciso, a série geométrica (&lt;a href=\"#seriepot\"&gt;4.1&lt;/a&gt;) é divergente no ponto\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\"&gt;&lt;/span&gt;. Mas como o teorema &lt;a href=\"#teointserie\"&gt;4.5&lt;/a&gt; afirma que a convergência pode ser alterada nos extremos do intervalo,\n        precisamos de uma nova investigação para nos certificarmos de que a série de interesse (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;), tornou-se\n        convergente em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\"&gt;&lt;/span&gt;. Note que para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\"&gt;&lt;/span&gt;, a série em (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;) torna-se,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\frac{1}{9} - \\frac{1}{11} + \\cdots = \\sum_{n=0}^{\\infty} (-1)^{n} \\frac{1}{2n+1}\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P31\" title=\"4P31\"&gt;\n        &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1683.svg\" alt=\"$\\displaystyle 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\frac{1}{9} - \\frac{1}{11} + \\cdots = \\sum_{n=0}^{\\infty} (-1)^{n} \\frac{1}{2n+1} $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P32\" style=\"text-indent: 0 !important;\" title=\"4P32\"&gt;\n        que é uma série alternada que satisfaz as condições do teorema &lt;a href=\"#seriealt\"&gt;4.4&lt;/a&gt; e portanto é convergente. A série\n        (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;) também converge se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1684.svg\" alt=\"$x = -1$\"&gt;&lt;/span&gt;, mas como este valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; está fora do nosso interesse, deixaremos os\n        detalhes para o leitor interessado.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P33\" title=\"4P33\"&gt;\n        Segue que a igualdade (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;) é válida também para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\"&gt;&lt;/span&gt; e como sabemos que &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1} 1 =\n        \\frac{\\pi}{4}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.76ex; vertical-align: -0.83ex; \" src=\"img/img1685.svg\" alt=\"${\\mathrm {tg}}^{-1} 1 =\n        \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;, então temos a fórmula,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{\\pi}{4} = {\\mathrm  {tg}}^{-1} 1 = 1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\frac{1}{9} - \\frac{1}{11} + \\cdots\n           = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{2n+1}, \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P34\" title=\"4P34\"&gt;\n        &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1686.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4} = {\\mathrm {tg}}^{-1} 1 = 1 - \\frac{1}{3} + \\frac{1...\n        ...c{1}{9} - \\frac{1}{11} + \\cdots\n        = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{2n+1}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P35\" style=\"text-indent: 0 !important;\" title=\"4P35\"&gt;\n        ou, ainda,\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P36\" title=\"4P36\"&gt;&lt;a id=\"piserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        \\pi = 4 - \\frac{4}{3} + \\frac{4}{5} - \\frac{4}{7} + \\frac{4}{9} - \\frac{4}{11} + \\cdots\n            = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}4}{2n+1}.\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1687.svg\" alt=\"$\\displaystyle \\pi = 4 - \\frac{4}{3} + \\frac{4}{5} - \\frac{4}{7} + \\frac{4}{9} - \\frac{4}{11} + \\cdots\n        = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}4}{2n+1}.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;3&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P37\" title=\"4P37\"&gt;\n        Esta expressão, obtida por volta de 1670, é conhecida como fórmula de Gregory-Leibniz. O problema desta fórmula é que a\n        convergência se dá de forma muito lenta, pois a série em (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;) converge para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img280.svg\" alt=\"$[-1,1]$\"&gt;&lt;/span&gt; e\n        como se pode ver, o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1304.svg\" alt=\"$x = 1$\"&gt;&lt;/span&gt; utilizado para obter a série está no extremo do intervalo. Quanto mais afastado do\n        centro deste intervalo, mais lenta a convergência. Não vamos discutir aqui os chamados “níveis de convergência” e\n        então para nós, convergência mais rápida significa obter mais casas decimais corretas com menos termos adicionados.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P38\" title=\"4P38\"&gt;\n        Vamos exemplificar o uso desta fórmula calculando uma aproximação para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; usando as 30 primeiras parcelas da soma\n        infinita.\n        &lt;/p&gt;\n        &lt;br /&gt;\n        &lt;a id=\"5610\"&gt;&lt;/a&gt;\n        &lt;table class=\"PAD  BORDER\"&gt;\n        &lt;caption&gt;&lt;strong&gt;Tabela 4.1:&lt;/strong&gt;\n        Aproximação de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; pela série de Gregory-Leibniz.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $(-1)^{n}\\dfrac{4}{2n+1}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1688.svg\" alt=\"$(-1)^{n}\\dfrac{4}{2n+1}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\approx \\, \\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.21ex; vertical-align: -0.10ex; \" src=\"img/img1689.svg\" alt=\"$\\approx \\, \\pi$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;0&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;4,0000000&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;4,0000000&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;1&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-1,3333333&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;2,6666667&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;2&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,8000000&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,4666667&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;3&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,5714286&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;2,8952381&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;4&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,4444444&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,3396825&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;5&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,3636364&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;2,9760461&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;6&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,3076923&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,2837384&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;7&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,2666667&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0170717&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;8&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,2352941&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,2523658&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;9&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,2105263&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0418395&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;10&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,1904762&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,2323157&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;11&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,1739130&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0584027&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;12&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,1600000&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,2184027&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;13&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,1481481&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0702546&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;14&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,1379310&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,2081856&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;15&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,1290323&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0791533&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;16&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,1212121&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,2003654&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;17&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,1142857&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0860797&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;18&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,1081081&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1941878&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;19&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,1025641&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0916237&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;20&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,0975610&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1891847&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;21&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,0930233&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0961614&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;22&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,0888889&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1850503&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;23&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,0851064&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,0999439&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;24&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,0816327&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1815766&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;25&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,0784314&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1031452&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;26&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,0754717&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1786169&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;27&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,0727273&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1058896&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;28&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;0,0701754&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1760650&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;29&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;-0,0677966&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1082684&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;a id=\"tabpi1\"&gt;&lt;/a&gt;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;/tr&gt;\n      \n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;br /&gt;\n        &lt;p class=\" unidade\" id=\"4P39\" title=\"4P39\"&gt;\n        \n        Note pela tabela acima, que a primeira casa decimal de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;, somente estabiliza-se quando já\n        foram somados 25 termos. Serão necessários 300 termos da série para que a segunda casa decimal seja igual a 4 e 5000\n        termos para obtermos a terceira casa decimal. Apesar disto, esta fórmula está longe de ser considerada inútil.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P40\" title=\"4P40\"&gt;\n        Algum tempo mais tarde, John Machin descobriu que a fórmula (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;) poderia ser usada para valores de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;\n        menores do que 1, obtendo assim, convergências mais rápidas. O problema é que não podemos simplesmente substituir &lt;!-- MATH\n         $x =\n        \\frac{1}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1690.svg\" alt=\"$x =\n        \\frac{1}{2}$\"&gt;&lt;/span&gt;, ou &lt;!-- MATH\n         $x = \\frac{1}{10}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1691.svg\" alt=\"$x = \\frac{1}{10}$\"&gt;&lt;/span&gt; em (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;) pois não conhecemos &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1}(\\frac{1}{2})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1692.svg\" alt=\"${\\mathrm {tg}}^{-1}(\\frac{1}{2})$\"&gt;&lt;/span&gt; ou\n        &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1}(\\frac{1}{10})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1693.svg\" alt=\"${\\mathrm {tg}}^{-1}(\\frac{1}{10})$\"&gt;&lt;/span&gt;. É necessário um argumento mais engenhoso.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P41\" title=\"4P41\"&gt;\n        Machin usou a fórmula da soma de arcos para a tangente\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}(u+v) = \\frac{{\\mathrm  {tg}}u + {\\mathrm  {tg}}v}{1- {\\mathrm  {tg}}u {\\mathrm  {tg}}v},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P42\" title=\"4P42\"&gt;\n        &lt;img style=\"height: 4.77ex; vertical-align: -2.03ex; \" src=\"img/img1694.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}(u+v) = \\frac{{\\mathrm {tg}}u + {\\mathrm {tg}}v}{1- {\\mathrm {tg}}u {\\mathrm {tg}}v}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P43\" style=\"text-indent: 0 !important;\" title=\"4P43\"&gt;\n        e obteve a identidade\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}\\left(x - \\frac{\\pi}{4}\\right) = \\frac{{\\mathrm  {tg}}x -1}{1 + {\\mathrm  {tg}}x},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P44\" title=\"4P44\"&gt;\n        &lt;img style=\"height: 5.00ex; vertical-align: -2.03ex; \" src=\"img/img1695.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}\\left(x - \\frac{\\pi}{4}\\right) = \\frac{{\\mathrm {tg}}x -1}{1 + {\\mathrm {tg}}x}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P45\" style=\"text-indent: 0 !important;\" title=\"4P45\"&gt;\n        e com &lt;!-- MATH\n         $x = 4{\\mathrm  {tg}}^{-1}(\\frac{1}{5})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1696.svg\" alt=\"$x = 4{\\mathrm {tg}}^{-1}(\\frac{1}{5})$\"&gt;&lt;/span&gt;, escreveu\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P46\" title=\"4P46\"&gt;&lt;a id=\"eqtg4atg\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        {\\mathrm  {tg}}\\left(4{\\mathrm  {tg}}^{-1}(\\tfrac{1}{5}) - \\frac{\\pi}{4}\\right)\n         = \\frac{{\\mathrm  {tg}}(4{\\mathrm  {tg}}^{-1}(\\frac{1}{5})) -1}{1 + {\\mathrm  {tg}}(4{\\mathrm  {tg}}^{-1}(\\frac{1}{5}))}.\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.06ex; vertical-align: -2.52ex; \" src=\"img/img1697.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}\\left(4{\\mathrm {tg}}^{-1}(\\tfrac{1}{5}) - \\frac{\\p...\n        ...^{-1}(\\frac{1}{5})) -1}{1 + {\\mathrm {tg}}(4{\\mathrm {tg}}^{-1}(\\frac{1}{5}))}.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;4&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P47\" title=\"4P47\"&gt;\n        Usando a fórmula da duplicação de arcos para a tangente, Machin calculou\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}(2{\\mathrm  {tg}}^{-1}(\\tfrac{1}{5})) = \\frac{2{\\mathrm  {tg}}({\\mathrm  {tg}}^{-1}(\\frac{1}{5}))}{1-{\\mathrm  {tg}}^{2}({\\mathrm  {tg}}^{-1}(\\frac{1}{5}))}\n         = \\frac{2 \\cdot \\frac{1}{5}}{1 - \\frac{1}{25}} = \\frac{5}{12}, \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P48\" title=\"4P48\"&gt;\n        &lt;img style=\"height: 6.06ex; vertical-align: -2.52ex; \" src=\"img/img1698.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}(2{\\mathrm {tg}}^{-1}(\\tfrac{1}{5})) = \\frac{2{\\mat...\n        ...frac{1}{5}))}\n        = \\frac{2 \\cdot \\frac{1}{5}}{1 - \\frac{1}{25}} = \\frac{5}{12}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P49\" style=\"text-indent: 0 !important;\" title=\"4P49\"&gt;\n        e depois\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}(4{\\mathrm  {tg}}^{-1}(\\tfrac{1}{5})) = \\frac{2{\\mathrm  {tg}}(2{\\mathrm  {tg}}^{-1}(\\frac{1}{5}))}{1-{\\mathrm  {tg}}^{2}(2{\\mathrm  {tg}}^{-1}(\\frac{1}{5}))}\n         = \\frac{2 \\cdot \\frac{5}{12}}{1-(\\frac{5}{12})^{2}} = \\frac{120}{119}, \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P50\" title=\"4P50\"&gt;\n        &lt;img style=\"height: 6.06ex; vertical-align: -2.52ex; \" src=\"img/img1699.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}(4{\\mathrm {tg}}^{-1}(\\tfrac{1}{5})) = \\frac{2{\\mat...\n        ...5}))}\n        = \\frac{2 \\cdot \\frac{5}{12}}{1-(\\frac{5}{12})^{2}} = \\frac{120}{119}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P51\" style=\"text-indent: 0 !important;\" title=\"4P51\"&gt;\n        que substituído em (&lt;a href=\"#eqtg4atg\"&gt;4.4&lt;/a&gt;) o levou a\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}\\left(4{\\mathrm  {tg}}^{-1}(\\tfrac{1}{5}) - \\frac{\\pi}{4}\\right) = \\frac{\\frac{120}{119}-1}{1+\\frac{120}{119}}\n         = \\frac{1}{239}. \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P52\" title=\"4P52\"&gt;\n        &lt;img style=\"height: 6.06ex; vertical-align: -2.52ex; \" src=\"img/img1700.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}\\left(4{\\mathrm {tg}}^{-1}(\\tfrac{1}{5}) - \\frac{\\pi}{4}\\right) = \\frac{\\frac{120}{119}-1}{1+\\frac{120}{119}}\n        = \\frac{1}{239}. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P53\" title=\"4P53\"&gt;\n        Aplicando arco tangente em ambos os membros e reorganizando os termos obtém-se a fórmula\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{\\pi}{4} = 4 {\\mathrm  {tg}}^{-1}(\\tfrac{1}{5}) - {\\mathrm  {tg}}^{-1}(\\tfrac{1}{239}),\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P54\" title=\"4P54\"&gt;\n        &lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1701.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4} = 4 {\\mathrm {tg}}^{-1}(\\tfrac{1}{5}) - {\\mathrm {tg}}^{-1}(\\tfrac{1}{239}), $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P55\" style=\"text-indent: 0 !important;\" title=\"4P55\"&gt;\n        que em 1706 foi usada por Machin para calcular 100 casas decimais para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P56\" title=\"4P56\"&gt;\n        A ideia de Machin, de reescrever &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1} 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img1702.svg\" alt=\"${\\mathrm {tg}}^{-1} 1$\"&gt;&lt;/span&gt; em somas de arco tangentes com argumentos menores, motivou outros\n        matemáticos. A igualdade\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P57\" title=\"4P57\"&gt;&lt;a id=\"idtgaddinv\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        {\\mathrm  {tg}}^{-1}( \\tfrac{1}{z} ) = {\\mathrm  {tg}}^{-1}( \\tfrac{1}{m} ) + {\\mathrm  {tg}}^{-1}( \\tfrac{1}{n} ),\n        \\end{equation}\n         --&gt;\n        &lt;table&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img1703.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}( \\tfrac{1}{z} ) = {\\mathrm {tg}}^{-1}( \\tfrac{1}{m} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{n} ),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;5&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"4P58\" style=\"text-indent: 0 !important;\" title=\"4P58\"&gt;\n        com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt; inteiros, se mostrou útil nesta abordagem. Note que se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt; são inteiros positivos, então\n        o fato de a função arco tangente ser crescente obrigará os valores de &lt;!-- MATH\n         $\\frac{1}{m}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1705.svg\" alt=\"$\\frac{1}{m}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $\\frac{1}{n}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1706.svg\" alt=\"$\\frac{1}{n}$\"&gt;&lt;/span&gt; serem menores do\n        que &lt;!-- MATH\n         $\\frac{1}{z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1707.svg\" alt=\"$\\frac{1}{z}$\"&gt;&lt;/span&gt;. Isto significa que &lt;!-- MATH\n         $\\frac{1}{m}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1705.svg\" alt=\"$\\frac{1}{m}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $\\frac{1}{n}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1706.svg\" alt=\"$\\frac{1}{n}$\"&gt;&lt;/span&gt; estarão mais próximos de 0 do que &lt;!-- MATH\n         $\\frac{1}{z}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1707.svg\" alt=\"$\\frac{1}{z}$\"&gt;&lt;/span&gt;, o\n        que torna a convergência mais rápida. Vamos primeiramente estabelecer qual a relação entre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt; para que a\n        identidade (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;) tenha sentido.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P59\" title=\"4P59\"&gt;\n        Aplicando a função tangente em ambos os membros de (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;), temos\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P60\" title=\"4P60\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1708.svg\" alt=\"$\\displaystyle \\frac{1}{z} = {\\mathrm {tg}}( {\\mathrm {tg}}^{-1}( \\tfrac{1}{z} ))$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1709.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}\\left( {\\mathrm {tg}}^{-1}( \\tfrac{1}{m} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{n} ) \\right)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.02ex; vertical-align: -2.49ex; \" src=\"img/img1710.svg\" alt=\"$\\displaystyle = \\frac{{\\mathrm {tg}}({\\mathrm {tg}}^{-1}(\\frac{1}{m})) + {\\math...\n        ...c{\\frac{1}{m} + \\frac{1}{n}}{1 - \\frac{1}{m} \\frac{1}{n}} = \\frac{n+m}{mn - 1},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"4P61\" style=\"text-indent: 0 !important;\" title=\"4P61\"&gt;\n        e portanto, obtemos que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\frac{1}{z} = \\frac{n+m}{mn - 1}.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P62\" title=\"4P62\"&gt;\n        &lt;img style=\"height: 4.69ex; vertical-align: -1.72ex; \" src=\"img/img1711.svg\" alt=\"$\\displaystyle \\frac{1}{z} = \\frac{n+m}{mn - 1}.$\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P63\" title=\"4P63\"&gt;\n        Desta igualdade, organizando os termos e somando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img1712.svg\" alt=\"$z^{2}$\"&gt;&lt;/span&gt; em ambos os membros, vem\n        &lt;!-- MATH\n         \\begin{displaymath}\n        mn - nz - mz + z^{2} = 1 + z^{2},\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P64\" title=\"4P64\"&gt;\n        &lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1713.svg\" alt=\"$\\displaystyle mn - nz - mz + z^{2} = 1 + z^{2}, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P65\" style=\"text-indent: 0 !important;\" title=\"4P65\"&gt;\n        ou ainda\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P66\" title=\"4P66\"&gt;&lt;a id=\"idmnz\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        (m - z)(n - z) = 1 + z^{2}.\n        \\end{equation}\n         --&gt;\n        &lt;table&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1714.svg\" alt=\"$\\displaystyle (m - z)(n - z) = 1 + z^{2}.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P67\" title=\"4P67\"&gt;\n        A igualdade (&lt;a href=\"#idmnz\"&gt;4.6&lt;/a&gt;) estabelece portanto uma relação entre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt;, para que a identidade\n        (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;) faça sentido. Como estamos interessados em desmembrar &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1} 1$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.58ex; \" src=\"img/img1702.svg\" alt=\"${\\mathrm {tg}}^{-1} 1$\"&gt;&lt;/span&gt; então faremos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.66ex; vertical-align: -0.11ex; \" src=\"img/img1715.svg\" alt=\"$z = 1$\"&gt;&lt;/span&gt; em\n        (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;) e (&lt;a href=\"#idmnz\"&gt;4.6&lt;/a&gt;), obtendo\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (m-1)(n-1) = 2.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P68\" title=\"4P68\"&gt;\n        &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1716.svg\" alt=\"$\\displaystyle (m-1)(n-1) = 2. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P69\" title=\"4P69\"&gt;\n        Basta então considerar &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1717.svg\" alt=\"$(m-1)$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1718.svg\" alt=\"$(n-1)$\"&gt;&lt;/span&gt; como sendo dois fatores inteiros do número 2. Escolhemos os fatores 1 e 2.\n        Colocando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1719.svg\" alt=\"$(m-1)=1$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1720.svg\" alt=\"$(n-1) = 2$\"&gt;&lt;/span&gt; obtemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1721.svg\" alt=\"$m = 2$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1722.svg\" alt=\"$n = 3$\"&gt;&lt;/span&gt; e, substituindo em (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;), temos a fórmula\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P70\" title=\"4P70\"&gt;&lt;a id=\"Eulerserie\"&gt;&lt;/a&gt;&lt;!-- MATH\n         \\begin{equation}\n        \\frac{\\pi}{4} = {\\mathrm  {tg}}^{-1} 1 = {\\mathrm  {tg}}^{-1}( \\tfrac{1}{2} ) + {\\mathrm  {tg}}^{-1}( \\tfrac{1}{3} ),\n        \\end{equation}\n         --&gt;\n        &lt;table &gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1723.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4} = {\\mathrm {tg}}^{-1} 1 = {\\mathrm {tg}}^{-1}( \\tfrac{1}{2} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} ),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        &lt;p class=\" unidade\" id=\"4P71\" style=\"text-indent: 0 !important;\" title=\"4P71\"&gt;\n        que foi obtida por Euler em 1738.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P72\" title=\"4P72\"&gt;\n        Observe que podemos novamente repetir esta ideia para modificar as arco tangentes das frações &lt;!-- MATH\n         $\\frac{1}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1724.svg\" alt=\"$\\frac{1}{2}$\"&gt;&lt;/span&gt; ou\n        &lt;!-- MATH\n         $\\frac{1}{3}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1725.svg\" alt=\"$\\frac{1}{3}$\"&gt;&lt;/span&gt; por outra soma de arco tangentes com argumentos menores ainda, para fazer convergências mais rápidas.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P73\" title=\"4P73\"&gt;\n        Este foi um método muito utilizado por matemáticos e várias fórmulas foram obtidas, conhecidas como fórmulas do tipo\n        Machin. Algumas delas são:\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P74\" title=\"4P74\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1727.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}^{-1}( \\tfrac{1}{2} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{5} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{8} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Strassnitzky)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1728.svg\" alt=\"$\\displaystyle = 2 {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{7} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Huton)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1729.svg\" alt=\"$\\displaystyle = 4 {\\mathrm {tg}}^{-1}(\\tfrac{1}{5}) - {\\mathrm {tg}}^{-1}(\\tfrac{1}{70}) + {\\mathrm {tg}}^{-1}(\\tfrac{1}{99})$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Euler, em 1764)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1730.svg\" alt=\"$\\displaystyle = 8 {\\mathrm {tg}}^{-1}( \\tfrac{1}{10} ) - {\\mathrm {tg}}^{-1}( \\tfrac{1}{239} ) - 4 {\\mathrm {tg}}^{-1}( \\tfrac{1}{515} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Klingenstierna)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1731.svg\" alt=\"$\\displaystyle = 12 {\\mathrm {tg}}^{-1}(\\tfrac{1}{18}) + 8 {\\mathrm {tg}}^{-1}(\\tfrac{1}{57}) - 5 {\\mathrm {tg}}^{-1}( \\tfrac{1}{239} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Gauss)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1732.svg\" alt=\"$\\displaystyle = 3 {\\mathrm {tg}}^{-1}(\\tfrac{1}{4}) + {\\mathrm {tg}}^{-1}(\\tfrac{1}{20}) + {\\mathrm {tg}}^{-1}(\\tfrac{1}{1985})$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Loney, em 1893)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.89ex; vertical-align: -0.86ex; \" src=\"img/img1733.svg\" alt=\"$\\displaystyle = 22 {\\mathrm {tg}}^{-1}(\\tfrac{1}{38}) + 17 {\\mathrm {tg}}^{-1}(\\tfrac{7}{601}) + 10 {\\mathrm {tg}}^{-1}(\\tfrac{7}{8149})$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Sebah)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.89ex; vertical-align: -0.86ex; \" src=\"img/img1734.svg\" alt=\"$\\displaystyle = 44 {\\mathrm {tg}}^{-1}(\\tfrac{1}{57}) + 7 {\\mathrm {tg}}^{-1}(\\...\n        ... {\\mathrm {tg}}^{-1}(\\tfrac{1}{682}) + 24 {\\mathrm {tg}}^{-1}(\\tfrac{1}{12943})$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Stormer, em 1896)&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1735.svg\" alt=\"$\\displaystyle = 12 {\\mathrm {tg}}^{-1}(\\tfrac{1}{49}) + 32 {\\mathrm {tg}}^{-1}(...\n        ...{\\mathrm {tg}}^{-1}(\\tfrac{1}{239}) + 12\n        {\\mathrm {tg}}^{-1}(\\tfrac{1}{110443})$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        (Takano, em 1982)&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P75\" title=\"4P75\"&gt;\n        A fórmula de Strassnitzky, é obtida a partir da fórmula de Euler, desmembrando o termo &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1}( \\frac{1}{3} )$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1736.svg\" alt=\"${\\mathrm {tg}}^{-1}( \\frac{1}{3} )$\"&gt;&lt;/span&gt;. Vamos\n        ver os detalhes. Considerando &lt;!-- MATH\n         $\\frac{1}{z} = \\frac{1}{3}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1737.svg\" alt=\"$\\frac{1}{z} = \\frac{1}{3}$\"&gt;&lt;/span&gt; e, portanto, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1738.svg\" alt=\"$z = 3$\"&gt;&lt;/span&gt;, em (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;) e (&lt;a href=\"#idmnz\"&gt;4.6&lt;/a&gt;),\n        obtemos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (m-3)(n-3) = 10.\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P76\" title=\"4P76\"&gt;\n        &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1739.svg\" alt=\"$\\displaystyle (m-3)(n-3) = 10. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P77\" title=\"4P77\"&gt;\n        Escolhemos agora dois fatores de 10. Considerando os fatores 2 e 5 e colocando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1740.svg\" alt=\"$(m-3)=2$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1741.svg\" alt=\"$(n-3)=5$\"&gt;&lt;/span&gt;, obtemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1742.svg\" alt=\"$m = 5$\"&gt;&lt;/span&gt;\n        e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1743.svg\" alt=\"$n = 8$\"&gt;&lt;/span&gt;. Temos portanto\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P78\" title=\"4P78\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1744.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}^{-1}( \\tfrac{1}{2} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1745.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}^{-1}( \\tfrac{1}{2} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{5} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{8} ).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P79\" title=\"4P79\"&gt;\n        A fórmula de Huton é obtida, também a partir da fórmula de Euler, desmembrando o termo &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1}( \\frac{1}{2} )$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1692.svg\" alt=\"${\\mathrm {tg}}^{-1}(\\frac{1}{2})$\"&gt;&lt;/span&gt;.\n        Considerando &lt;!-- MATH\n         $\\frac{1}{z} = \\frac{1}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1746.svg\" alt=\"$\\frac{1}{z} = \\frac{1}{2}$\"&gt;&lt;/span&gt;, isto é, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1747.svg\" alt=\"$z=2$\"&gt;&lt;/span&gt; e, substituindo em (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;) e (&lt;a href=\"#idmnz\"&gt;4.6&lt;/a&gt;), temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        (m-2)(n-2) = 5,\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P80\" title=\"4P80\"&gt;\n        &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1748.svg\" alt=\"$\\displaystyle (m-2)(n-2) = 5, $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P81\" style=\"text-indent: 0 !important;\" title=\"4P81\"&gt;\n        e escolhendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1749.svg\" alt=\"$(m-2) = 1$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1750.svg\" alt=\"$(n-2) = 5$\"&gt;&lt;/span&gt; temos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1751.svg\" alt=\"$m = 3$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1752.svg\" alt=\"$n = 7$\"&gt;&lt;/span&gt; e com estes valores\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P82\" title=\"4P82\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1744.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}^{-1}( \\tfrac{1}{2} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1753.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} ) + {\\mathrm {tg}}^{-1}( \\tfr...\n        ... = 2 {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} ) +\n        {\\mathrm {tg}}^{-1}( \\tfrac{1}{7} ).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P83\" title=\"4P83\"&gt;\n        Levando em conta ainda que podemos considerar que os fatores, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt;, do número &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img1754.svg\" alt=\"$z^{2}+1$\"&gt;&lt;/span&gt; sejam inteiros negativos,\n        e usando o fato de que arco tangente é uma função ímpar, isto é, &lt;!-- MATH\n         ${\\mathrm  {tg}}^{-1}(-\\frac{1}{n}) = - {\\mathrm  {tg}}^{-1}(\\frac{1}{n})$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1755.svg\" alt=\"${\\mathrm {tg}}^{-1}(-\\frac{1}{n}) = - {\\mathrm {tg}}^{-1}(\\frac{1}{n})$\"&gt;&lt;/span&gt;,\n        conseguimos o cancelamento de termos em algumas substituições. Mais ainda, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt; não precisam ser números\n        inteiros, já que o desenvolvimento aplicado em (&lt;a href=\"#idtgaddinv\"&gt;4.5&lt;/a&gt;) é válido para quaisquer argumentos reais no domínio\n        da função arco tangente.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P84\" title=\"4P84\"&gt;\n        Podemos verificar que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}^{-1}(\\tfrac{1}{3}) = {\\mathrm  {tg}}^{-1}(\\tfrac{1}{7}) + {\\mathrm  {tg}}^{-1}(\\tfrac{2}{11}),\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P85\" title=\"4P85\"&gt;\n        &lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1756.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}(\\tfrac{1}{3}) = {\\mathrm {tg}}^{-1}(\\tfrac{1}{7}) + {\\mathrm {tg}}^{-1}(\\tfrac{2}{11}), $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P86\" style=\"text-indent: 0 !important;\" title=\"4P86\"&gt;\n        e que\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}^{-1}(\\tfrac{2}{11}) = {\\mathrm  {tg}}^{-1}(\\tfrac{1}{7}) + {\\mathrm  {tg}}^{-1}(\\tfrac{3}{79}),\n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P87\" title=\"4P87\"&gt;\n        &lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1757.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}(\\tfrac{2}{11}) = {\\mathrm {tg}}^{-1}(\\tfrac{1}{7}) + {\\mathrm {tg}}^{-1}(\\tfrac{3}{79}), $\"&gt;\n        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P88\" style=\"text-indent: 0 !important;\" title=\"4P88\"&gt;\n        e substituindo estas duas igualdades na fórmula de Huton, obtém-se\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P89\" title=\"4P89\"&gt;&lt;table class=\"equation\"&gt;\n        &lt;tbody&gt;&lt;tr&gt;\n        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.04ex; vertical-align: -1.56ex; \" src=\"img/img1726.svg\" alt=\"$\\displaystyle \\frac{\\pi}{4}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1728.svg\" alt=\"$\\displaystyle = 2 {\\mathrm {tg}}^{-1}( \\tfrac{1}{3} ) + {\\mathrm {tg}}^{-1}( \\tfrac{1}{7} )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.86ex; vertical-align: -0.83ex; \" src=\"img/img1758.svg\" alt=\"$\\displaystyle = 3 {\\mathrm {tg}}^{-1}(\\tfrac{1}{7}) + 2{\\mathrm {tg}}^{-1}(\\tfrac{2}{11})$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;tr&gt;\n        &lt;td&gt;&nbsp;&lt;/td&gt;\n        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.87ex; vertical-align: -0.85ex; \" src=\"img/img1759.svg\" alt=\"$\\displaystyle = 5 {\\mathrm {tg}}^{-1}(\\tfrac{1}{7}) + {\\mathrm {tg}}^{-1}(\\tfrac{3}{79}).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n        \n        \n        &lt;p class=\" unidade\" id=\"4P90\" title=\"4P90\"&gt;\n        Vamos comparar os resultados obtidos na tabela &lt;a href=\"#tabpi1\"&gt;4.1&lt;/a&gt;, calculando agora uma aproximação de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; pela série de\n        Euler (&lt;a href=\"#Eulerserie\"&gt;4.7&lt;/a&gt;). Fazendo &lt;!-- MATH\n         $x = \\frac{1}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1690.svg\" alt=\"$x =\n        \\frac{1}{2}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $x = \\frac{1}{3}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1760.svg\" alt=\"$x = \\frac{1}{3}$\"&gt;&lt;/span&gt; em (&lt;a href=\"#arctanserie\"&gt;4.2&lt;/a&gt;), temos\n        &lt;!-- MATH\n         \\begin{displaymath}\n        {\\mathrm  {tg}}^{-1}(\\tfrac{1}{2}) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{2^{2n+1}(2n+1)}, \\qquad \\text{e} \\qquad\n           {\\mathrm  {tg}}^{-1}(\\tfrac{1}{3}) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{3^{2n+1}(2n+1)}. \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P91\" title=\"4P91\"&gt;\n        &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1761.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}(\\tfrac{1}{2}) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{2^{2n+1}(2n+1)},$\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1762.svg\" alt=\"$\\displaystyle \\qquad\n        {\\mathrm {tg}}^{-1}(\\tfrac{1}{3}) = \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{3^{2n+1}(2n+1)}. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P92\" title=\"4P92\"&gt;\n        Então temos que,\n        &lt;!-- MATH\n         \\begin{displaymath}\n        \\pi = 4{\\mathrm  {tg}}^{-1}(\\tfrac{1}{2}) + 4{\\mathrm  {tg}}^{-1}(\\tfrac{1}{3})\n           = \\sum_{n=0}^{\\infty} (-1)^{n} \\frac{4}{2^{2n+1}(2n+1)} + \\sum_{n=0}^{\\infty} (-1)^{n} \\frac{4}{3^{2n+1}(2n+1)}. \n        \\end{displaymath}\n         --&gt;\n        &lt;/p&gt;\n        &lt;div class=\"mathdisplay unidade\" id=\"4P93\" title=\"4P93\"&gt;\n        &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1763.svg\" alt=\"$\\displaystyle \\pi = 4{\\mathrm {tg}}^{-1}(\\tfrac{1}{2}) + 4{\\mathrm {tg}}^{-1}(\\...\n        ...ac{4}{2^{2n+1}(2n+1)} + \\sum_{n=0}^{\\infty} (-1)^{n} \\frac{4}{3^{2n+1}(2n+1)}. $\"&gt;\n        &lt;/div&gt;\n        \n        &lt;p class=\" unidade\" id=\"4P94\" title=\"4P94\"&gt;\n        Abaixo segue uma tabela de convergência para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; com os 15 primeiros termos desta última série.\n        &lt;/p&gt;\n        &lt;br /&gt;\n        &lt;a id=\"6070\"&gt;&lt;/a&gt;\n        &lt;table class=\"PAD  BORDER\"&gt;\n        &lt;caption&gt;&lt;strong&gt;Tabela 4.2:&lt;/strong&gt;\n        Aproximação de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; pela série de Euler.&lt;/caption&gt;\n        &lt;tbody&gt;&lt;tr&gt;&lt;td class=\"CENTER\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;&lt;!-- MATH\n         $S_{1} = \\frac{(-1)^{n}4}{2^{2n+1}(2n+1)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.60ex; vertical-align: -1.32ex; \" src=\"img/img1764.svg\" alt=\"$S_{1} = \\frac{(-1)^{n}4}{2^{2n+1}(2n+1)}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;&lt;!-- MATH\n         $S_{2} = \\frac{(-1)^{n}4}{3^{2n+1}(2n+1)}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.60ex; vertical-align: -1.32ex; \" src=\"img/img1765.svg\" alt=\"$S_{2} = \\frac{(-1)^{n}4}{3^{2n+1}(2n+1)}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;&lt;!-- MATH\n         $S_{1} + S_{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img1766.svg\" alt=\"$S_{1} + S_{2}$\"&gt;&lt;/span&gt; &nbsp;&nbsp;&nbsp;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&lt;!-- MATH\n         $\\approx \\pi$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.21ex; vertical-align: -0.10ex; \" src=\"img/img1767.svg\" alt=\"$\\approx \\pi$\"&gt;&lt;/span&gt;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;&nbsp;&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;0&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;2,0000000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;1,3333333333&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;3,3333333333&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,3333333333&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;1&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,1666666667&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0493827160&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,2160493827&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1172839506&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;2&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0250000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0032921811&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0282921811&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1455761317&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;3&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0044642857&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0002612842&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0047255699&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1408505618&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;4&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0008680556&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000225801&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0008906357&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1417411974&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;5&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0001775568&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000020527&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0001796096&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415615879&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;6&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000375601&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000001930&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000377531&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415993410&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;7&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000081380&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000186&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000081566&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415911844&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;8&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000017952&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000018&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000017970&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415929813&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;9&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000004015&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000002&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000004017&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415925796&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;10&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000908&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000908&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415926705&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;11&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000207&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000207&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415926497&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;12&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000048&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000048&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415926545&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;13&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000011&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;-0,0000000011&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415926534&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;14&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000003&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000000&lt;/td&gt;\n        &lt;td class=\"RIGHT\"&gt;0,0000000003&lt;/td&gt;\n        &lt;td class=\"CENTER\"&gt;3,1415926536&lt;/td&gt;\n        &lt;/tr&gt;\n        &lt;tr&gt;&lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n            &lt;td class=\"RIGHT\"&gt;&nbsp;&lt;/td&gt;\n            &lt;td class=\"RIGHT\"&gt;&nbsp;&lt;/td&gt;\n            &lt;td class=\"RIGHT\"&gt;&nbsp;&lt;/td&gt;\n            &lt;td class=\"CENTER\"&gt;&nbsp;&lt;/td&gt;\n            &lt;/tr&gt;\n        &lt;/tbody&gt;&lt;/table&gt;\n        &lt;br /&gt;\n        &lt;p class=\" unidade\" id=\"4P95\" title=\"4P95\"&gt;\n        Observe que esta série converge muito mais rápido do que a série (&lt;a href=\"#piserie\"&gt;4.3&lt;/a&gt;). Com apenas 15\n        termos somados, temos 9 casas decimais corretas de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;. Note ainda que na terceira coluna, os valores vão para zero\n        mais rápido do que na segunda coluna. Como dissemos antes, isto ocorre pois a terceira coluna representa os valores da\n        série arco tangente de &lt;!-- MATH\n         $\\frac{1}{3}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1725.svg\" alt=\"$\\frac{1}{3}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n         $\\frac{1}{3}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.85ex; \" src=\"img/img1725.svg\" alt=\"$\\frac{1}{3}$\"&gt;&lt;/span&gt; está mais próximo de 0 do que &lt;!-- MATH\n         $\\frac{1}{2}$\n         --&gt;\n        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1724.svg\" alt=\"$\\frac{1}{2}$\"&gt;&lt;/span&gt;.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P96\" title=\"4P96\"&gt;\n        Atualmente o trabalho de calcular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; é feito com o auxílio de supercomputadores, que trabalham por horas ou até dias\n        para calcular trilhões de casas decimais. As fórmulas baseadas em arco tangente são bastante utilizadas por\n        apresentarem apenas números racionais.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P97\" title=\"4P97\"&gt;\n        A questão principal é por que calcular &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; com trilhões de casas decimais? Sabe-se que umas poucas casas decimais\n        resolvem todos os problemas práticos de engenharia, física ou matemática. Para ser mais preciso, 39 casas decimais\n        permitem calcular a medida da circunferência do universo com erro menor do que o diâmetro de um átomo de hidrogênio.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P98\" title=\"4P98\"&gt;\n        Uma aplicação prática é o teste de microprocessadores. Quando um computador ou um processador numérico é desenvolvido,\n        é necessário saber até que ponto sua eficiência numérica é confiável, e nestes termos, nada melhor do que testá-lo a\n        calcular um número já conhecido. Calcular dígitos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;, já não é mais uma questão de conhecer este número, mas sim\n        de comprovar o poder dos computadores.\n        &lt;/p&gt;\n        &lt;p class=\" unidade\" id=\"4P99\" title=\"4P99\"&gt;\n        Para uma coleção maior de fórmulas envolvendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; recomendamos [&lt;a href=\"/trigonometria-hiperbolica/referencias#Sebah\"&gt;6&lt;/a&gt;, Sebah], [&lt;a href=\"/trigonometria-hiperbolica/referencias#Eymard\"&gt;1&lt;/a&gt;, Eymard] e também\n        [&lt;a href=\"/trigonometria-hiperbolica/referencias#Weisstein\"&gt;9&lt;/a&gt;, Weisstein]. Comentários e demonstrações sobre outras fórmulas para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt; podem ser encontrados em\n        [&lt;a href=\"/trigonometria-hiperbolica/referencias#Eymard\"&gt;1&lt;/a&gt;, Eymard].\n        &lt;/p&gt;",
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    "section": "4.2 Cálculo de integrais",
    "text": "4.2 Cálculo de integrais\n\n        &lt;a id=\"secinttrig\"&gt;&lt;/a&gt;\n\n\n            &lt;p class=\" unidade\" id=\"4P100\" title=\"4P100\"&gt;\n            Dentre as aplicações clássicas e imediatas do Cálculo Diferencial e Integral estão o cálculo de áreas de regiões e de\n            comprimentos de curvas determinadas por funções. Estes cálculos em geral reduzem-se ao cálculo de integrais envolvendo\n            tais funções. Entretanto, determinar certas integrais não é tarefa tão fácil. Existem várias regras de integração porém\n            muitas funções não se enquadram nas técnicas tradicionais de integração.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P101\" title=\"4P101\"&gt;\n            Dentre várias situações podemos citar como exemplo\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P102\" title=\"4P102\"&gt;&lt;a id=\"integrais\"&gt;&lt;/a&gt;&lt;!-- MATH\n             \\begin{equation}\n            \\int \\sqrt{r^{2}-x^{2}}dx \\qquad \\text{e} \\qquad \\int \\sqrt{r^{2}+x^{2}} dx,\n            \\end{equation}\n             --&gt;\n            &lt;table&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1768.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}-x^{2}}dx$\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1769.svg\" alt=\"$\\displaystyle \\qquad \\int \\sqrt{r^{2}+x^{2}} dx,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;8&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"4P103\" style=\"text-indent: 0 !important;\" title=\"4P103\"&gt;\n            para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1123.svg\" alt=\"$r&gt;0$\"&gt;&lt;/span&gt; uma constante e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; no devido intervalo de definição das funções consideradas.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P104\" title=\"4P104\"&gt;\n            A primeira integral aparece, por exemplo, no cálculo de áreas de regiões circulares. Mais precisamente, dado &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1123.svg\" alt=\"$r&gt;0$\"&gt;&lt;/span&gt;, a\n            área &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\"&gt;&lt;/span&gt; compreendida entre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; e o gráfico do semi-círculo &lt;!-- MATH\n             $f(x) = \\sqrt{r^{2}-x^{2}}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img1770.svg\" alt=\"$f(x) = \\sqrt{r^{2}-x^{2}}$\"&gt;&lt;/span&gt;, no intervalo &lt;!-- MATH\n             $[a,b]\n            \\subset [-r,r]$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1771.svg\" alt=\"$[a,b]\n            \\subset [-r,r]$\"&gt;&lt;/span&gt;, é calculada pela integral definida\n            &lt;!-- MATH\n             \\begin{displaymath}\n            A = \\int_{a}^{b} \\sqrt{r^{2}-x^{2}} dx.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P105\" title=\"4P105\"&gt;\n            &lt;img style=\"height: 5.42ex; vertical-align: -2.01ex; \" src=\"img/img1772.svg\" alt=\"$\\displaystyle A = \\int_{a}^{b} \\sqrt{r^{2}-x^{2}} dx. $\"&gt;\n            &lt;/div&gt;\n            \n            &lt;div class=\"CENTER\"&gt;&lt;a id=\"6102\"&gt;&lt;/a&gt;\n            &lt;table&gt;\n            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 4.1:&lt;/strong&gt;\n             Área sob a semicircunferência. &lt;/caption&gt;\n            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n            &lt;div class=\"CENTER\"&gt;\n            &lt;img src=\"img/areaint.png\" alt=\"Image areaint\"&gt;\n            &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"4P106\" title=\"4P106\"&gt;\n            A segunda integral em (&lt;a href=\"#integrais\"&gt;4.8&lt;/a&gt;) aparece, por exemplo, no cálculo de comprimento de curvas. O comprimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt; da\n            curva determinada pelo gráfico de uma função quadrática &lt;!-- MATH\n             $g(x) = kx^{2}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1773.svg\" alt=\"$g(x) = kx^{2}$\"&gt;&lt;/span&gt;, para alguma constante &lt;!-- MATH\n             $k \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img289.svg\" alt=\"$k \\in \\mathbb{R}$\"&gt;&lt;/span&gt;, no intervalo\n            &lt;!-- MATH\n             $[a,b] \\subset \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1774.svg\" alt=\"$[a,b] \\subset \\mathbb{R}$\"&gt;&lt;/span&gt;, é dado por\n            &lt;!-- MATH\n             \\begin{displaymath}\n            C = \\int_{a}^{b} \\sqrt{1+[g'(x)]^{2}} dx = \\frac{1}{r} \\int_{a}^{b} \\sqrt{r^{2}+x^{2}} dx,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P107\" title=\"4P107\"&gt;\n            &lt;img style=\"height: 5.42ex; vertical-align: -2.01ex; \" src=\"img/img1775.svg\" alt=\"$\\displaystyle C = \\int_{a}^{b} \\sqrt{1+[g'(x)]^{2}} dx = \\frac{1}{r} \\int_{a}^{b} \\sqrt{r^{2}+x^{2}} dx, $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P108\" style=\"text-indent: 0 !important;\" title=\"4P108\"&gt;\n            sendo que &lt;!-- MATH\n             $r = \\frac{1}{2k}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1776.svg\" alt=\"$r = \\frac{1}{2k}$\"&gt;&lt;/span&gt; tornou-se uma constante de ajuste.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P109\" title=\"4P109\"&gt;\n            O aluno de um curso de Cálculo Diferencial e Integral, quando se depara com alguma das integrais em (&lt;a href=\"#integrais\"&gt;4.8&lt;/a&gt;),\n            recorre às fórmulas de integração prontas que geralmente figuram nas últimas páginas dos livros. Mas estas fórmulas não\n            “caíram do céu”. Nesta seção vamos mostrar como as funções trigonométricas podem ajudar a determinar as integrais em\n            (&lt;a href=\"#integrais\"&gt;4.8&lt;/a&gt;). Não vamos nos preocupar com o intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1777.svg\" alt=\"$[a,b]$\"&gt;&lt;/span&gt; e então faremos os cálculos considerando as\n            integrais indefinidas.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P110\" title=\"4P110\"&gt;\n            Para a primeira integral em (&lt;a href=\"#integrais\"&gt;4.8&lt;/a&gt;), precisaremos antes o cálculo auxiliar da integral\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\cos^{2}x dx.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P111\" title=\"4P111\"&gt;\n            &lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1778.svg\" alt=\"$\\displaystyle \\int \\cos^{2}x dx. $\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"4P112\" title=\"4P112\"&gt;\n            Usando as identidades\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P113\" title=\"4P113\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1779.svg\" alt=\"$\\displaystyle \\cos^{2}x + {\\mathrm {sen}}^{2}x = 1,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1780.svg\" alt=\"$\\displaystyle \\cos^{2}x - {\\mathrm {sen}}^{2}x = \\cos(2x),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"4P114\" style=\"text-indent: 0 !important;\" title=\"4P114\"&gt;\n            obtemos\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\cos^{2}x = \\tfrac{1}{2} + \\tfrac{1}{2} \\cos(2x),\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P115\" title=\"4P115\"&gt;\n            &lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img1781.svg\" alt=\"$\\displaystyle \\cos^{2}x = \\tfrac{1}{2} + \\tfrac{1}{2} \\cos(2x), $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P116\" style=\"text-indent: 0 !important;\" title=\"4P116\"&gt;\n            e, também,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            {\\mathrm {sen}}^{2}x = \\tfrac{1}{2} - \\tfrac{1}{2} \\cos(2x).\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P117\" title=\"4P117\"&gt;\n            &lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img1782.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}^{2}x = \\tfrac{1}{2} - \\tfrac{1}{2} \\cos(2x). $\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"4P118\" title=\"4P118\"&gt;\n            Integrando temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P119\" title=\"4P119\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1783.svg\" alt=\"$\\displaystyle \\int \\cos^{2}x dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.82ex; vertical-align: -0.83ex; \" src=\"img/img1784.svg\" alt=\"$\\displaystyle = \\tfrac{1}{2}x + \\tfrac{1}{4} {\\mathrm {sen}}(2x),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1785.svg\" alt=\"$\\displaystyle \\int {\\mathrm {sen}}^{2}x dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.82ex; vertical-align: -0.83ex; \" src=\"img/img1786.svg\" alt=\"$\\displaystyle = \\tfrac{1}{2}x - \\tfrac{1}{4} {\\mathrm {sen}}(2x).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P120\" title=\"4P120\"&gt;\n            O leitor atento diria agora que esquecemos a constante de integração nas duas expressões acima. Por simplicidade,\n            durante os cálculos omitiremos a constante de integração, que só será apresentada ao final para não perder\n            definitivamente o rigor matemático. Agora vamos ao cálculo da primeira integral em (&lt;a href=\"#integrais\"&gt;4.8&lt;/a&gt;). Considerando que\n            &lt;!-- MATH\n             $x \\in [-r,r]$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1787.svg\" alt=\"$x \\in [-r,r]$\"&gt;&lt;/span&gt;, queremos determinar\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\sqrt{r^{2}-x^{2}}dx.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P121\" title=\"4P121\"&gt;\n            &lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1788.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}-x^{2}}dx. $\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"4P122\" title=\"4P122\"&gt;\n            Fazendo a mudança de variáveis &lt;!-- MATH\n             $x = r{\\mathrm {sen}}t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.55ex; vertical-align: -0.13ex; \" src=\"img/img1789.svg\" alt=\"$x = r{\\mathrm {sen}}t$\"&gt;&lt;/span&gt;, temos que &lt;!-- MATH\n             $\\frac{dx}{dt} = r \\cos t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1790.svg\" alt=\"$\\frac{dx}{dt} = r \\cos t$\"&gt;&lt;/span&gt; e, substituindo na integral, temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P123\" title=\"4P123\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1768.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}-x^{2}}dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1791.svg\" alt=\"$\\displaystyle = \\int \\sqrt{r^{2}-(r{\\mathrm {sen}}t)^{2}} \\,\\, r \\cos t dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1792.svg\" alt=\"$\\displaystyle = \\int r\\sqrt{1- {\\mathrm {sen}}^{2} t} \\,\\, r \\cos t dt = r^{2} \\int \\sqrt{\\cos^{2} t} \\cos t dt.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P124\" title=\"4P124\"&gt;\n            Lembremos agora que &lt;!-- MATH\n             $x \\in [-r,r]$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1787.svg\" alt=\"$x \\in [-r,r]$\"&gt;&lt;/span&gt; e então a substituição &lt;!-- MATH\n             $x = r {\\mathrm {sen}}t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.55ex; vertical-align: -0.13ex; \" src=\"img/img1789.svg\" alt=\"$x = r{\\mathrm {sen}}t$\"&gt;&lt;/span&gt; obriga (bijetivamente) &lt;!-- MATH\n             $t \\in\n            [-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1793.svg\" alt=\"$t \\in\n            [-\\frac{\\pi}{2}, \\frac{\\pi}{2}]$\"&gt;&lt;/span&gt;. Neste intervalo temos &lt;!-- MATH\n             $\\cos t &gt; 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1794.svg\" alt=\"$\\cos t &gt; 0$\"&gt;&lt;/span&gt; e então\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P125\" title=\"4P125\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1768.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}-x^{2}}dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1795.svg\" alt=\"$\\displaystyle = r^{2} \\int \\sqrt{\\cos^{2} t} \\cos t dt = r^{2} \\int \\cos t \\cos t dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.26ex; vertical-align: -1.92ex; \" src=\"img/img1796.svg\" alt=\"$\\displaystyle = r^{2} \\int \\cos^{2} t dt = r^{2} (\\tfrac{1}{2}t + \\tfrac{1}{4} {\\mathrm {sen}}(2t)) = \\frac{r^{2}}{2}t + \\frac{r^{2}}{2} {\\mathrm {sen}}t \\cos t.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P126\" title=\"4P126\"&gt;\n            Voltando à variável original &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, temos que &lt;!-- MATH\n             ${\\mathrm {sen}}t = \\frac{x}{r}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img1797.svg\" alt=\"${\\mathrm {sen}}t = \\frac{x}{r}$\"&gt;&lt;/span&gt;, ou ainda &lt;!-- MATH\n             $t = {\\mathrm {sen}}^{-1}(\\frac{x}{r})$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.75ex; vertical-align: -0.83ex; \" src=\"img/img1798.svg\" alt=\"$t = {\\mathrm {sen}}^{-1}(\\frac{x}{r})$\"&gt;&lt;/span&gt; e, assim,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P127\" title=\"4P127\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1768.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}-x^{2}}dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1799.svg\" alt=\"$\\displaystyle = \\frac{r^{2}}{2}t + \\frac{r^{2}}{2} {\\mathrm {sen}}t \\cos t$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.33ex; vertical-align: -1.60ex; \" src=\"img/img1800.svg\" alt=\"$\\displaystyle = \\frac{r^{2}}{2} {\\mathrm {sen}}^{-1}(\\tfrac{x}{r}) + \\frac{r^{2...\n            ...^{2}}{2} {\\mathrm {sen}}^{-1}(\\tfrac{x}{r}) + \\frac{1}{2} x\n            \\sqrt{r^{2}-x^{2}}.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P128\" title=\"4P128\"&gt;\n            Obtemos então a fórmula de integração presente nos livros de Cálculo Diferencial e Integral\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\sqrt{r^{2}-x^{2}}dx = \\frac{r^{2}}{2} {\\mathrm {sen}}^{-1}(\\tfrac{x}{r}) + \\frac{x}{2} \\sqrt{r^{2}-x^{2}} + C,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P129\" title=\"4P129\"&gt;\n            &lt;img style=\"height: 5.26ex; vertical-align: -1.92ex; \" src=\"img/img1801.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}-x^{2}}dx = \\frac{r^{2}}{2} {\\mathrm {sen}}^{-1}(\\tfrac{x}{r}) + \\frac{x}{2} \\sqrt{r^{2}-x^{2}} + C, $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P130\" style=\"text-indent: 0 !important;\" title=\"4P130\"&gt;\n            para alguma constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P131\" title=\"4P131\"&gt;\n            A segunda integral em (&lt;a href=\"#integrais\"&gt;4.8&lt;/a&gt;) é obtida de forma análoga porém com funções trigonométricas hiperbólicas.\n            Primeiro vamos determinar a integral\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\cosh^{2}x dx.\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P132\" title=\"4P132\"&gt;\n            &lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1802.svg\" alt=\"$\\displaystyle \\int \\cosh^{2}x dx. $\"&gt;\n            &lt;/div&gt;\n            \n            &lt;p class=\" unidade\" id=\"4P133\" title=\"4P133\"&gt;\n            Usando as identidades\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P134\" title=\"4P134\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img1803.svg\" alt=\"$\\displaystyle \\cosh^{2}x - {\\mathrm{senh}}^{2}x = 1,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1804.svg\" alt=\"$\\displaystyle \\cosh^{2}x + {\\mathrm{senh}}^{2}x = \\cosh(2x),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"4P135\" style=\"text-indent: 0 !important;\" title=\"4P135\"&gt;\n            temos que\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P136\" title=\"4P136\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img1805.svg\" alt=\"$\\displaystyle \\cosh^{2}x = \\tfrac{1}{2} + \\tfrac{1}{2} \\cosh(2x),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.85ex; vertical-align: -0.83ex; \" src=\"img/img1806.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}^{2}x = \\tfrac{1}{2} \\cosh(2x) - \\tfrac{1}{2},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"4P137\" style=\"text-indent: 0 !important;\" title=\"4P137\"&gt;\n            e integrando\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P138\" title=\"4P138\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1807.svg\" alt=\"$\\displaystyle \\int \\cosh^{2}x dx = \\tfrac{1}{4} {\\mathrm{senh}}(2x) + \\tfrac{1}{2}x,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1808.svg\" alt=\"$\\displaystyle \\int {\\mathrm{senh}}^{2}x dx = \\tfrac{1}{4} {\\mathrm{senh}}(2x) - \\tfrac{1}{2}x.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P139\" title=\"4P139\"&gt;\n            Vamos agora determinar a integral de interesse,\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\sqrt{r^{2}+x^{2}} dx,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P140\" title=\"4P140\"&gt;\n            &lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1809.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}+x^{2}} dx, $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P141\" style=\"text-indent: 0 !important;\" title=\"4P141\"&gt;\n            considerando &lt;!-- MATH\n             $x \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1151.svg\" alt=\"$x \\in \\mathbb{R}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1123.svg\" alt=\"$r&gt;0$\"&gt;&lt;/span&gt; uma constante arbitrária. Fazendo a mudança de variáveis &lt;!-- MATH\n             $x = r {\\mathrm{senh}}t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1810.svg\" alt=\"$x = r {\\mathrm{senh}}t$\"&gt;&lt;/span&gt; temos que\n            &lt;!-- MATH\n             $\\frac{dx}{dt} = r \\cosh t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1811.svg\" alt=\"$\\frac{dx}{dt} = r \\cosh t$\"&gt;&lt;/span&gt; e então temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P142\" title=\"4P142\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1812.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}+x^{2}} dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1813.svg\" alt=\"$\\displaystyle = \\int \\sqrt{r^{2}+(r {\\mathrm{senh}}t)^{2}} \\,\\, r \\cosh t dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1814.svg\" alt=\"$\\displaystyle = \\int r \\sqrt{1+{\\mathrm{senh}}^{2} t} \\,\\, r \\cosh t dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1815.svg\" alt=\"$\\displaystyle = r^{2} \\int \\sqrt{1+{\\mathrm{senh}}^{2} t} \\,\\, \\cosh t dt = r^{2} \\int \\sqrt{\\cosh^{2} t} \\,\\, \\cosh t dt.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P143\" title=\"4P143\"&gt;\n            Lembrando agora que a função seno hiperbólico é bijetiva de &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\"&gt;&lt;/span&gt; em &lt;!-- MATH\n             $\\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\"&gt;&lt;/span&gt; e então a mudança de variável &lt;!-- MATH\n             $x = r {\\mathrm{senh}}t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1810.svg\" alt=\"$x = r {\\mathrm{senh}}t$\"&gt;&lt;/span&gt;\n            permite &lt;!-- MATH\n             $t \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img688.svg\" alt=\"$t \\in \\mathbb{R}$\"&gt;&lt;/span&gt; e como &lt;!-- MATH\n             $\\cosh t &gt; 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1816.svg\" alt=\"$\\cosh t &gt; 0$\"&gt;&lt;/span&gt; para qualquer &lt;!-- MATH\n             $t \\in \\mathbb{R}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img688.svg\" alt=\"$t \\in \\mathbb{R}$\"&gt;&lt;/span&gt;, temos\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P144\" title=\"4P144\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1812.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}+x^{2}} dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1817.svg\" alt=\"$\\displaystyle = r^{2} \\int \\sqrt{\\cosh^{2} t} \\,\\, \\cosh t dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1818.svg\" alt=\"$\\displaystyle = r^{2} \\int \\cosh t \\,\\, \\cosh t dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1819.svg\" alt=\"$\\displaystyle = r^{2}( \\tfrac{1}{4} {\\mathrm{senh}}(2t) + \\tfrac{1}{2}t) = \\frac{r^{2}}{2} {\\mathrm{senh}}t \\cosh t + \\frac{r^{2}}{2}t,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            &lt;p class=\" unidade\" id=\"4P145\" style=\"text-indent: 0 !important;\" title=\"4P145\"&gt;\n            e voltando à variável &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, temos &lt;!-- MATH\n             ${\\mathrm{senh}}t = \\frac{x}{r}$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img1820.svg\" alt=\"${\\mathrm{senh}}t = \\frac{x}{r}$\"&gt;&lt;/span&gt;, ou ainda &lt;!-- MATH\n             $t = {\\mathrm{senh}}^{-1}(\\frac{x}{r})$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.83ex; vertical-align: -0.83ex; \" src=\"img/img1821.svg\" alt=\"$t = {\\mathrm{senh}}^{-1}(\\frac{x}{r})$\"&gt;&lt;/span&gt; e, assim,\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P146\" title=\"4P146\"&gt;&lt;table class=\"equation\"&gt;\n            &lt;tbody&gt;&lt;tr&gt;\n            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img1812.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}+x^{2}} dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.89ex; vertical-align: -1.55ex; \" src=\"img/img1822.svg\" alt=\"$\\displaystyle = \\frac{r^{2}}{2} {\\mathrm{senh}}t \\cosh t + \\frac{r^{2}}{2}t$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;tr&gt;\n            &lt;td&gt;&nbsp;&lt;/td&gt;\n            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.33ex; vertical-align: -1.60ex; \" src=\"img/img1823.svg\" alt=\"$\\displaystyle = \\frac{r^{2}}{2} \\frac{x}{r} \\sqrt{1+\\frac{x^{2}}{r^{2}}} + \\fra...\n            ...{x}{2} \\sqrt{r^{2}+x^{2}} + \\frac{r^{2}}{2} {\\mathrm{senh}}^{-1}(\\tfrac{x}{r}).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n            \n            \n            &lt;p class=\" unidade\" id=\"4P147\" title=\"4P147\"&gt;\n            Neste caso, podemos escrever esta igualdade sem o uso explícito da função arco seno hiperbólico. Usando a identidade\n            &lt;!-- MATH\n             ${\\mathrm{senh}}^{-1} x = \\ln(x + \\sqrt{x^{2} + 1})$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.70ex; vertical-align: -0.62ex; \" src=\"img/img1824.svg\" alt=\"${\\mathrm{senh}}^{-1} x = \\ln(x + \\sqrt{x^{2} + 1})$\"&gt;&lt;/span&gt;, obtida na seção &lt;a href=\"/trigonometria-hiperbolica/igualdades-exponenciais-e-logaritmicas#secforlog\"&gt;3.3&lt;/a&gt;, escrevemos\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\sqrt{r^{2}+x^{2}} dx = \\frac{x}{2} \\sqrt{r^{2}+x^{2}} + \\frac{r^{2}}{2} \\ln\\left( \\tfrac{x}{r} + \\sqrt{\\tfrac{x^{2}}{r^{2}} + 1} \\right),\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P148\" title=\"4P148\"&gt;\n            &lt;img style=\"height: 5.45ex; vertical-align: -2.10ex; \" src=\"img/img1825.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}+x^{2}} dx = \\frac{x}{2} \\sqrt{r^{2}+x^{2}} + \\frac{r^{2}}{2} \\ln\\left( \\tfrac{x}{r} + \\sqrt{\\tfrac{x^{2}}{r^{2}} + 1} \\right), $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P149\" style=\"text-indent: 0 !important;\" title=\"4P149\"&gt;\n            e, reorganizando os termos temos, finalmente\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\sqrt{r^{2}+x^{2}} dx = \\frac{x}{2} \\sqrt{r^{2}+x^{2}} + \\frac{r^{2}}{2} \\ln\\left( x + \\sqrt{x^{2} + r^{2}} \\right) + C,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P150\" title=\"4P150\"&gt;\n            &lt;img style=\"height: 5.26ex; vertical-align: -1.92ex; \" src=\"img/img1826.svg\" alt=\"$\\displaystyle \\int \\sqrt{r^{2}+x^{2}} dx = \\frac{x}{2} \\sqrt{r^{2}+x^{2}} + \\frac{r^{2}}{2} \\ln\\left( x + \\sqrt{x^{2} + r^{2}} \\right) + C, $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P151\" style=\"text-indent: 0 !important;\" title=\"4P151\"&gt;\n            sendo que a constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt; absorve o termo constante &lt;!-- MATH\n             $-\\frac{r^{2}}{2} \\ln r$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.09ex; vertical-align: -0.83ex; \" src=\"img/img1827.svg\" alt=\"$-\\frac{r^{2}}{2} \\ln r$\"&gt;&lt;/span&gt; que desprezamos na\n            reorganização dos termos.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P152\" title=\"4P152\"&gt;\n            A fórmula de integração\n            &lt;!-- MATH\n             \\begin{displaymath}\n            \\int \\sqrt{x^{2}-r^{2}} dx = \\frac{x}{2} \\sqrt{x^{2}-r^{2}} - \\frac{r^{2}}{2} \\ln\\left( x + \\sqrt{x^{2} - r^{2}} \\right) + C,\n            \\end{displaymath}\n             --&gt;\n            &lt;/p&gt;\n            &lt;div class=\"mathdisplay unidade\" id=\"4P153\" title=\"4P153\"&gt;\n            &lt;img style=\"height: 5.26ex; vertical-align: -1.92ex; \" src=\"img/img1828.svg\" alt=\"$\\displaystyle \\int \\sqrt{x^{2}-r^{2}} dx = \\frac{x}{2} \\sqrt{x^{2}-r^{2}} - \\frac{r^{2}}{2} \\ln\\left( x + \\sqrt{x^{2} - r^{2}} \\right) + C, $\"&gt;\n            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P154\" style=\"text-indent: 0 !important;\" title=\"4P154\"&gt;\n            válida para &lt;!-- MATH\n             $x \\geq r &gt; 0$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img1829.svg\" alt=\"$x \\geq r &gt; 0$\"&gt;&lt;/span&gt;, é obtida de maneira análoga pela substituição &lt;!-- MATH\n             $x = r \\cosh t$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1830.svg\" alt=\"$x = r \\cosh t$\"&gt;&lt;/span&gt;. Deixamos agora os detalhes\n            para o leitor.\n            &lt;/p&gt;\n            &lt;p class=\" unidade\" id=\"4P155\" title=\"4P155\"&gt;\n            Podemos ainda observar que em algumas situações, o cálculo da integral &lt;!-- MATH\n             $\\int \\sqrt{r^{2}-x^{2}}dx$\n             --&gt;\n            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.82ex; vertical-align: -0.74ex; \" src=\"img/img1831.svg\" alt=\"$\\int \\sqrt{r^{2}-x^{2}}dx$\"&gt;&lt;/span&gt; não é efetuado em\n            coordenadas cartesianas. Como é sabido, as coordenadas cartesianas dificultam o cálculo de integrais em regiões\n            circulares. Neste tipo de região é recomendado o uso de coordenadas polares. Mas a conversão de coordenadas cartesianas\n            para polares (e vice-versa) faz uso das funções trigonométricas também.\n            &lt;/p&gt;",
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    "title": "Capítulo 4: Aplicações",
    "section": "4.3 A catenária",
    "text": "4.3 A catenária\n\n                &lt;p class=\" unidade\" id=\"4P156\" title=\"4P156\"&gt;\n                Catenária é o nome da curva que descreve a trajetória de equilíbrio de um cabo flexível, de comprimento fixo e suspenso\n                por duas hastes. O estudo desta curva desempenha um papel fundamental nos cursos de engenharia.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"4P157\" title=\"4P157\"&gt;\n                Consideremos então um cabo flexível, sustentado por duas hastes, pelos pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img53.svg\" alt=\"$B$\"&gt;&lt;/span&gt;. Fixemos um sistema coordenado\n                cartesiano com o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1832.svg\" alt=\"$Ox$\"&gt;&lt;/span&gt; no nível do solo e o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1833.svg\" alt=\"$Oy$\"&gt;&lt;/span&gt; perpendicular ao solo passando pelo ponto mais baixo do cabo.\n                O cabo descreve uma curva neste sistema coordenado. Denotemos por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1201.svg\" alt=\"$y = f(x)$\"&gt;&lt;/span&gt; esta curva. Chamemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1834.svg\" alt=\"$C = (0,c)$\"&gt;&lt;/span&gt; o ponto\n                mais baixo da curva, que está sobre o eixo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img1833.svg\" alt=\"$Oy$\"&gt;&lt;/span&gt;. Tomemos um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1835.svg\" alt=\"$P = (x,y)$\"&gt;&lt;/span&gt; sobre esta curva e sem perda de\n                generalidade, consideremos o ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\"&gt;&lt;/span&gt; à direita de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt; no sistema coordenado considerado, isto é, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1835.svg\" alt=\"$P = (x,y)$\"&gt;&lt;/span&gt; com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img637.svg\" alt=\"$x&gt;0$\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"4P158\" title=\"4P158\"&gt;\n                Considerando a porção do cabo entre os pontos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\"&gt;&lt;/span&gt;, temos as forças &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img1836.svg\" alt=\"$\\vec{h}$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img1837.svg\" alt=\"$\\vec{p}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt;, atuando\n                sobre esta porção do cabo. &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img1837.svg\" alt=\"$\\vec{p}$\"&gt;&lt;/span&gt; é a força peso, que é decomposta nas componentes horizontal e vertical por\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\vec{p} = (0, -\\omega L)\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P159\" title=\"4P159\"&gt;\n                &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1839.svg\" alt=\"$\\displaystyle \\vec{p} = (0, -\\omega L) $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P160\" style=\"text-indent: 0 !important;\" title=\"4P160\"&gt;\n                sendo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1840.svg\" alt=\"$\\omega$\"&gt;&lt;/span&gt; é o peso do cabo por unidade de comprimento e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1841.svg\" alt=\"$L$\"&gt;&lt;/span&gt; é o comprimento do cabo (da porção do cabo\n                considerada). &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt; é a força de tração pela direita no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1835.svg\" alt=\"$P = (x,y)$\"&gt;&lt;/span&gt; e é decomposta nas componentes horizontal\n                e vertical por\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\vec{t} = (t \\cos \\theta, t {\\mathrm {sen}}\\theta),\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P161\" title=\"4P161\"&gt;\n                &lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img1842.svg\" alt=\"$\\displaystyle \\vec{t} = (t \\cos \\theta, t {\\mathrm {sen}}\\theta), $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P162\" style=\"text-indent: 0 !important;\" title=\"4P162\"&gt;\n                sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt; o módulo da tensão pela direita e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt; o ângulo que o vetor tangencial &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt; faz com a horizontal.\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img1836.svg\" alt=\"$\\vec{h}$\"&gt;&lt;/span&gt; é a força de tração pela esquerda no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1834.svg\" alt=\"$C = (0,c)$\"&gt;&lt;/span&gt;, dada por\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\vec{h} = (-h,0),\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P163\" title=\"4P163\"&gt;\n                &lt;img style=\"height: 2.77ex; vertical-align: -0.62ex; \" src=\"img/img1844.svg\" alt=\"$\\displaystyle \\vec{h} = (-h,0), $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P164\" style=\"text-indent: 0 !important;\" title=\"4P164\"&gt;\n                sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img293.svg\" alt=\"$h$\"&gt;&lt;/span&gt; o módulo da tensão pela esquerda.\n                &lt;/p&gt;\n                \n                \n                &lt;div class=\"CENTER\"&gt;&lt;a id=\"6377\"&gt;&lt;/a&gt;\n                &lt;table&gt;\n                &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 4.2:&lt;/strong&gt;\n                Forças atuantes no cabo suspenso.&lt;/caption&gt;\n                &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                &lt;div class=\"CENTER\"&gt;\n                &lt;img src=\"img/catenaria.png\" alt=\"Image catenaria\"&gt;\n                &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P165\" title=\"4P165\"&gt;\n                O sistema está em equilíbrio, isto é,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\vec{h} + \\vec{p} + \\vec{t} = \\vec{0},\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P166\" title=\"4P166\"&gt;\n                &lt;img style=\"height: 2.65ex; vertical-align: -0.50ex; \" src=\"img/img1845.svg\" alt=\"$\\displaystyle \\vec{h} + \\vec{p} + \\vec{t} = \\vec{0}, $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P167\" style=\"text-indent: 0 !important;\" title=\"4P167\"&gt;\n                e então\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                (-h,0) + (0,-\\omega L) + (t \\cos \\theta, t {\\mathrm {sen}}\\theta) = (0,0),\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P168\" title=\"4P168\"&gt;\n                &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1846.svg\" alt=\"$\\displaystyle (-h,0) + (0,-\\omega L) + (t \\cos \\theta, t {\\mathrm {sen}}\\theta) = (0,0), $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P169\" style=\"text-indent: 0 !important;\" title=\"4P169\"&gt;\n                donde temos\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P170\" title=\"4P170\"&gt;&lt;table class=\"equation\"&gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1847.svg\" alt=\"$\\displaystyle -h + t \\cos \\theta = 0$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;tr&gt;\n                &lt;td&gt;&nbsp;&lt;/td&gt;\n                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1848.svg\" alt=\"$\\displaystyle -\\omega L + t {\\mathrm {sen}}\\theta = 0.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"4P171\" title=\"4P171\"&gt;\n                Agora, sabemos do cálculo diferencial e integral que a inclinação &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt;, do vetor tangente à curva em um ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;,\n                se relaciona com a curva por\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                {\\mathrm  {tg}}\\theta = y',\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P172\" title=\"4P172\"&gt;\n                &lt;img style=\"height: 2.40ex; vertical-align: -0.58ex; \" src=\"img/img1849.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}\\theta = y', $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P173\" style=\"text-indent: 0 !important;\" title=\"4P173\"&gt;\n                donde temos que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                y' = {\\mathrm  {tg}}\\theta = \\frac{{\\mathrm {sen}}\\theta}{\\cos \\theta} = \\frac{\\frac{\\omega L}{t}}{\\frac{h}{t}} = \\frac{\\omega L}{h}.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P174\" title=\"4P174\"&gt;\n                &lt;img style=\"height: 6.02ex; vertical-align: -2.49ex; \" src=\"img/img1850.svg\" alt=\"$\\displaystyle y' = {\\mathrm {tg}}\\theta = \\frac{{\\mathrm {sen}}\\theta}{\\cos \\theta} = \\frac{\\frac{\\omega L}{t}}{\\frac{h}{t}} = \\frac{\\omega L}{h}. $\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P175\" title=\"4P175\"&gt;\n                Mas note que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1841.svg\" alt=\"$L$\"&gt;&lt;/span&gt; não é uma constante. &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1851.svg\" alt=\"$L = L(x)$\"&gt;&lt;/span&gt; é o comprimento da curva de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt; a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img4.svg\" alt=\"$P$\"&gt;&lt;/span&gt; e isto dependerá da posição do\n                ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1835.svg\" alt=\"$P = (x,y)$\"&gt;&lt;/span&gt;. Sabemos (do cálculo) que o comprimento desta curva pode ser calculado pela fórmula integral,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                L = L(x) = \\int_{0}^{x} \\sqrt{1+(y')^{2}} \\,\\, dx,\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P176\" title=\"4P176\"&gt;\n                &lt;img style=\"height: 5.08ex; vertical-align: -2.03ex; \" src=\"img/img1852.svg\" alt=\"$\\displaystyle L = L(x) = \\int_{0}^{x} \\sqrt{1+(y')^{2}} \\,\\, dx, $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P177\" style=\"text-indent: 0 !important;\" title=\"4P177\"&gt;\n                e, assim,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                y' = \\frac{\\omega L}{h} = \\frac{\\omega}{h} \\int_{0}^{x} \\sqrt{1+(y')^{2}} \\,\\, dx.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P178\" title=\"4P178\"&gt;\n                &lt;img style=\"height: 5.08ex; vertical-align: -2.03ex; \" src=\"img/img1853.svg\" alt=\"$\\displaystyle y' = \\frac{\\omega L}{h} = \\frac{\\omega}{h} \\int_{0}^{x} \\sqrt{1+(y')^{2}} \\,\\, dx. $\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P179\" title=\"4P179\"&gt;\n                Para eliminar a integral do segundo membro, derivamos ambos os membros da igualdade e, usando o Teorema Fundamental do\n                Cálculo, obtemos\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P180\" title=\"4P180\"&gt;&lt;a id=\"catenaria\"&gt;&lt;/a&gt;&lt;!-- MATH\n                 \\begin{equation}\n                y'' = \\frac{\\omega}{h} \\sqrt{1+(y')^{2}}.\n                \\end{equation}\n                 --&gt;\n                &lt;table &gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.39ex; vertical-align: -1.55ex; \" src=\"img/img1854.svg\" alt=\"$\\displaystyle y'' = \\frac{\\omega}{h} \\sqrt{1+(y')^{2}}.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                    (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;9&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                \n                \n                &lt;p class=\" unidade\" id=\"4P181\" title=\"4P181\"&gt;\n                A esta equação diferencial, juntamos as condições iniciais &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img1855.svg\" alt=\"$y'(0) = 0$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1856.svg\" alt=\"$y(0) = c$\"&gt;&lt;/span&gt;. Com o intuito de encontrar\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1201.svg\" alt=\"$y = f(x)$\"&gt;&lt;/span&gt;, a função que descreve a curva catenária, vamos resolver esta equação diferencial. Fazendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img1857.svg\" alt=\"$z=y'$\"&gt;&lt;/span&gt; e\n                substituindo em (&lt;a href=\"#catenaria\"&gt;4.9&lt;/a&gt;), temos a equação diferencial de ordem 1,\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                z' = \\tfrac{\\omega}{h} \\sqrt{1+z^{2}},\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P182\" title=\"4P182\"&gt;\n                &lt;img style=\"height: 3.21ex; vertical-align: -0.83ex; \" src=\"img/img1858.svg\" alt=\"$\\displaystyle z' = \\tfrac{\\omega}{h} \\sqrt{1+z^{2}}, $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P183\" style=\"text-indent: 0 !important;\" title=\"4P183\"&gt;\n                que pode ser reescrita na forma\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\frac{z'}{\\sqrt{1+z^{2}}} = \\frac{\\omega}{h}.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P184\" title=\"4P184\"&gt;\n                &lt;img style=\"height: 5.19ex; vertical-align: -2.06ex; \" src=\"img/img1859.svg\" alt=\"$\\displaystyle \\frac{z'}{\\sqrt{1+z^{2}}} = \\frac{\\omega}{h}. $\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P185\" title=\"4P185\"&gt;\n                Integrando em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, obtemos\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P186\" title=\"4P186\"&gt;&lt;!-- MATH\n                 \\begin{equation}\n                \\int \\frac{z'}{\\sqrt{1+z^{2}}} dx = \\frac{\\omega}{h} x + k,\n                \\end{equation}\n                 --&gt;\n                &lt;table &gt;\n                &lt;tbody&gt;&lt;tr&gt;\n                &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.19ex; vertical-align: -2.06ex; \" src=\"img/img1860.svg\" alt=\"$\\displaystyle \\int \\frac{z'}{\\sqrt{1+z^{2}}} dx = \\frac{\\omega}{h} x + k,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                    (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;10&lt;/span&gt;)  &lt;/td&gt;&lt;/tr&gt;\n                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                &lt;p class=\" unidade\" id=\"4P187\" style=\"text-indent: 0 !important;\" title=\"4P187\"&gt;\n                para alguma constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\"&gt;&lt;/span&gt; que ainda será determinada. Para determinar a integral do primeiro membro,\n                notemos que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                \\int \\frac{z'}{\\sqrt{1+z^{2}}} dx = \\int \\frac{\\frac{dz}{dx}}{\\sqrt{1+z^{2}}}dx = \\int \\frac{1}{\\sqrt{1+z^{2}}} dz.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P188\" title=\"4P188\"&gt;\n                &lt;img style=\"height: 5.59ex; vertical-align: -2.06ex; \" src=\"img/img1861.svg\" alt=\"$\\displaystyle \\int \\frac{z'}{\\sqrt{1+z^{2}}} dx = \\int \\frac{\\frac{dz}{dx}}{\\sqrt{1+z^{2}}}dx = \\int \\frac{1}{\\sqrt{1+z^{2}}} dz. $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P189\" style=\"text-indent: 0 !important;\" title=\"4P189\"&gt;\n                e levando em conta que a fração &lt;!-- MATH\n                 $\\frac{1}{\\sqrt{1+z^{2}}}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.29ex; vertical-align: -1.31ex; \" src=\"img/img1862.svg\" alt=\"$\\frac{1}{\\sqrt{1+z^{2}}}$\"&gt;&lt;/span&gt; é a derivada da função &lt;!-- MATH\n                 ${\\mathrm{senh}}^{-1} z$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.14ex; vertical-align: -0.13ex; \" src=\"img/img1863.svg\" alt=\"${\\mathrm{senh}}^{-1} z$\"&gt;&lt;/span&gt; (em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1412.svg\" alt=\"$z$\"&gt;&lt;/span&gt;),\n                segue que\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                {\\mathrm{senh}}^{-1} z = \\int \\frac{1}{\\sqrt{1+z^{2}}} dz = \\frac{\\omega}{h}x + k,\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P190\" title=\"4P190\"&gt;\n                &lt;img style=\"height: 5.03ex; vertical-align: -2.06ex; \" src=\"img/img1864.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}^{-1} z = \\int \\frac{1}{\\sqrt{1+z^{2}}} dz = \\frac{\\omega}{h}x + k, $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P191\" style=\"text-indent: 0 !important;\" title=\"4P191\"&gt;\n                para alguma constante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\"&gt;&lt;/span&gt;.\n                &lt;/p&gt;\n                &lt;p class=\" unidade\" id=\"4P192\" title=\"4P192\"&gt;\n                Substituindo a condição inicial &lt;!-- MATH\n                 $z(0) = y'(0) = 0$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img1865.svg\" alt=\"$z(0) = y'(0) = 0$\"&gt;&lt;/span&gt;, obtemos o valor da constante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img226.svg\" alt=\"$k = 0$\"&gt;&lt;/span&gt;. Voltando para a variável &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\"&gt;&lt;/span&gt;,\n                e com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img226.svg\" alt=\"$k = 0$\"&gt;&lt;/span&gt;, obtemos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                {\\mathrm{senh}}^{-1}(y') = \\frac{\\omega}{h} x.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P193\" title=\"4P193\"&gt;\n                &lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img1866.svg\" alt=\"$\\displaystyle {\\mathrm{senh}}^{-1}(y') = \\frac{\\omega}{h} x. $\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P194\" title=\"4P194\"&gt;\n                Aplicando a função seno hiperbólico em ambos os membros, temos\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                y' = {\\mathrm{senh}}(\\tfrac{\\omega}{h}x),\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P195\" title=\"4P195\"&gt;\n                &lt;img style=\"height: 2.64ex; vertical-align: -0.83ex; \" src=\"img/img1867.svg\" alt=\"$\\displaystyle y' = {\\mathrm{senh}}(\\tfrac{\\omega}{h}x), $\"&gt;\n                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P196\" style=\"text-indent: 0 !important;\" title=\"4P196\"&gt;\n                e integrando ambos os membros em relação a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, vem\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                y = \\frac{h}{\\omega} \\cosh(\\tfrac{\\omega}{h} x) + k.\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P197\" title=\"4P197\"&gt;\n                &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1868.svg\" alt=\"$\\displaystyle y = \\frac{h}{\\omega} \\cosh(\\tfrac{\\omega}{h} x) + k. $\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P198\" title=\"4P198\"&gt;\n                Usando a condição inicial &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1856.svg\" alt=\"$y(0) = c$\"&gt;&lt;/span&gt;, conseguimos o valor da nova constante de integração &lt;!-- MATH\n                 $k = (c-\\frac{h}{\\omega})$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1869.svg\" alt=\"$k = (c-\\frac{h}{\\omega})$\"&gt;&lt;/span&gt;.\n                Segue portanto a função procurada\n                &lt;!-- MATH\n                 \\begin{displaymath}\n                y = \\frac{h}{\\omega} \\cosh(\\tfrac{\\omega}{h} x) + \\left( c-\\tfrac{h}{\\omega} \\right).\n                \\end{displaymath}\n                 --&gt;\n                &lt;/p&gt;\n                &lt;div class=\"mathdisplay unidade\" id=\"4P199\" title=\"4P199\"&gt;\n                &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1870.svg\" alt=\"$\\displaystyle y = \\frac{h}{\\omega} \\cosh(\\tfrac{\\omega}{h} x) + \\left( c-\\tfrac{h}{\\omega} \\right). $\"&gt;\n                &lt;/div&gt;\n                \n                &lt;p class=\" unidade\" id=\"4P200\" title=\"4P200\"&gt;\n                Desta forma, obtemos que a curva catenária é descrita por um cosseno hiperbólico. O termo de translação\n                &lt;!-- MATH\n                 $(c-\\frac{h}{\\omega})$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1871.svg\" alt=\"$(c-\\frac{h}{\\omega})$\"&gt;&lt;/span&gt; pode ser manipulado mudando-se a origem do sistema coordenado fixado sobre a curva catenária. Os\n                fatores &lt;!-- MATH\n                 $\\frac{h}{\\omega}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1872.svg\" alt=\"$\\frac{h}{\\omega}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                 $\\frac{\\omega}{h}$\n                 --&gt;\n                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img1873.svg\" alt=\"$\\frac{\\omega}{h}$\"&gt;&lt;/span&gt; determinam a abertura da curva.\n                &lt;/p&gt;",
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    "section": "4.4 Série de Fourier",
    "text": "4.4 Série de Fourier\n\n                    &lt;p class=\" unidade\" id=\"4P201\" title=\"4P201\"&gt;\n                    O objetivo desta seção é apresentar as séries de Fourier. Jean Baptiste Fourier (1768-1830) foi o primeiro matemático a\n                    investigar séries envolvendo senos e cossenos e, por isso, essas séries levam hoje o seu nome. Ele introduziu esse\n                    assunto em 1822 em seu livro &lt;i&gt;Théorie Analytique de la Chaleur&lt;/i&gt; (Teoria analítica do calor).\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P202\" title=\"4P202\"&gt;\n                    Consideremos uma barra de comprimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1841.svg\" alt=\"$L$\"&gt;&lt;/span&gt;, com extremos em contato com um material de temperatura constante igual a\n                    zero. Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1874.svg\" alt=\"$u(x,t)$\"&gt;&lt;/span&gt; é a temperatura desta barra no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; e no instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; é a distribuição inicial da\n                    temperatura da barra e o fluxo de calor na extremidade da barra é proporcional à temperatura da extremidade, então a\n                    função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1874.svg\" alt=\"$u(x,t)$\"&gt;&lt;/span&gt; satisfaz as equações\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\left\\{ \\begin{array}{l} u_{t} = \\alpha^{2} u_{xx} \\quad \\text{em} \\quad (0,L) \\times (0,\\infty), \\\\\n                                                u(0,t) + u_{x}(0,t) = u(L,t) + u_{x}(L,t) = 0 \\quad \\text{para} \\quad t \\geq 0, \\\\\n                                                u(x,0) = f(x) \\quad \\text{para} \\quad x \\in [0,L], \\end{array} \\right. \n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P203\" title=\"4P203\"&gt;\n                    &lt;img style=\"height: 9.60ex; vertical-align: -4.29ex; \" src=\"img/img1875.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l} u_{t} = \\alpha^{2} u_{xx} \\quad \\text{em...\n                    ..., \\\\\n                    u(x,0) = f(x) \\quad \\text{para} \\quad x \\in [0,L], \\end{array} \\right. $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P204\" style=\"text-indent: 0 !important;\" title=\"4P204\"&gt;\n                    sendo que &lt;!-- MATH\n                     $u_{t} = \\frac{\\partial u}{\\partial t}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1876.svg\" alt=\"$u_{t} = \\frac{\\partial u}{\\partial t}$\"&gt;&lt;/span&gt;, &lt;!-- MATH\n                     $u_{x} = \\frac{\\partial u}{\\partial x}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1877.svg\" alt=\"$u_{x} = \\frac{\\partial u}{\\partial x}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                     $u_{xx} =\n                    \\frac{\\partial^{2} u}{\\partial x^{2}}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.20ex; vertical-align: -0.94ex; \" src=\"img/img1878.svg\" alt=\"$u_{xx} =\n                    \\frac{\\partial^{2} u}{\\partial x^{2}}$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P205\" title=\"4P205\"&gt;\n                    Esta é uma equação diferencial parcial, sujeita às condições de contorno e condições iniciais. Vamos resolver esta\n                    equação. O método que usaremos é conhecido como método das variáveis separáveis.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P206\" title=\"4P206\"&gt;\n                    Este método consiste em supor que a função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1874.svg\" alt=\"$u(x,t)$\"&gt;&lt;/span&gt; possa ser expressa como um produto de duas funções, uma dependendo\n                    de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt; e outra dependendo de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, isto é, supomos que &lt;!-- MATH\n                     $u(x,t) = \\varphi(x) \\eta(t)$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1879.svg\" alt=\"$u(x,t) = \\varphi(x) \\eta(t)$\"&gt;&lt;/span&gt;. Substituindo na equação\n                    diferencial parcial, temos\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\varphi(x) \\eta'(t) = \\alpha^{2} \\eta(t) \\varphi''(x),\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P207\" title=\"4P207\"&gt;\n                    &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1880.svg\" alt=\"$\\displaystyle \\varphi(x) \\eta'(t) = \\alpha^{2} \\eta(t) \\varphi''(x), $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P208\" style=\"text-indent: 0 !important;\" title=\"4P208\"&gt;\n                    ou, ainda,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{1}{\\alpha^{2}} \\frac{\\eta'(t)}{\\eta(t)} = \\frac{\\varphi''(x)}{\\varphi(x)}.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P209\" title=\"4P209\"&gt;\n                    &lt;img style=\"height: 5.20ex; vertical-align: -2.07ex; \" src=\"img/img1881.svg\" alt=\"$\\displaystyle \\frac{1}{\\alpha^{2}} \\frac{\\eta'(t)}{\\eta(t)} = \\frac{\\varphi''(x)}{\\varphi(x)}. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P210\" title=\"4P210\"&gt;\n                    Observe que o membro da direita não depende de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, enquanto o membro da esquerda não depende de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;. Isto sugere que\n                    na verdade ambos os membros não dependam nem de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; e nem de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, isto é, são constantes. Fisicamente esta constante é\n                    negativa considerando que a taxa de variação da temperatura está diminuindo. Por uma questão de facilidade no\n                    desenvolvimento dos cálculos, esta constante negativa é escrita como &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1882.svg\" alt=\"$-\\lambda$\"&gt;&lt;/span&gt;, com &lt;!-- MATH\n                     $\\lambda &gt; 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1883.svg\" alt=\"$\\lambda &gt; 0$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P211\" title=\"4P211\"&gt;\n                    Temos então\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\frac{1}{\\alpha^{2}} \\frac{\\eta'(t)}{\\eta(t)} = -\\lambda \\qquad \\text{e} \\qquad \\frac{\\varphi''(x)}{\\varphi(x)} = -\\lambda,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P212\" title=\"4P212\"&gt;\n                    &lt;img style=\"height: 5.20ex; vertical-align: -2.07ex; \" src=\"img/img1884.svg\" alt=\"$\\displaystyle \\frac{1}{\\alpha^{2}} \\frac{\\eta'(t)}{\\eta(t)} = -\\lambda$\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 5.20ex; vertical-align: -2.07ex; \" src=\"img/img1885.svg\" alt=\"$\\displaystyle \\qquad \\frac{\\varphi''(x)}{\\varphi(x)} = -\\lambda, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P213\" style=\"text-indent: 0 !important;\" title=\"4P213\"&gt;\n                    ou ainda,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\eta'(t) + \\lambda\\alpha^{2} \\eta(t) = 0 \\qquad \\text{e} \\qquad \\varphi''(x) + \\lambda \\varphi(x) = 0.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P214\" title=\"4P214\"&gt;\n                    &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img1886.svg\" alt=\"$\\displaystyle \\eta'(t) + \\lambda\\alpha^{2} \\eta(t) = 0$\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img1887.svg\" alt=\"$\\displaystyle \\qquad \\varphi''(x) + \\lambda \\varphi(x) = 0. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P215\" title=\"4P215\"&gt;\n                    Temos agora duas equações diferenciais ordinárias, lineares e homogêneas. As soluções são fáceis de serem obtidas e\n                    são, precisamente,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P216\" title=\"4P216\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1888.svg\" alt=\"$\\displaystyle \\varphi(x)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.78ex; vertical-align: -0.62ex; \" src=\"img/img1889.svg\" alt=\"$\\displaystyle = a\\cos(\\sqrt{\\lambda}x) + b{\\mathrm {sen}}(\\sqrt{\\lambda}x)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1890.svg\" alt=\"$\\displaystyle \\eta(t)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.40ex; vertical-align: -0.10ex; \" src=\"img/img1891.svg\" alt=\"$\\displaystyle = ce^{-\\lambda \\alpha^{2} t}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P217\" style=\"text-indent: 0 !important;\" title=\"4P217\"&gt;\n                    para quaisquer coeficientes &lt;!-- MATH\n                     $a,b,c \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1892.svg\" alt=\"$a,b,c \\in \\mathbb{R}$\"&gt;&lt;/span&gt;. Segue que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    u(x,t) = \\left( a\\cos(\\sqrt{\\lambda}x) + b{\\mathrm {sen}}(\\sqrt{\\lambda}x) \\right) ce^{-\\lambda \\alpha^{2} t}.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P218\" title=\"4P218\"&gt;\n                    &lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img1893.svg\" alt=\"$\\displaystyle u(x,t) = \\left( a\\cos(\\sqrt{\\lambda}x) + b{\\mathrm {sen}}(\\sqrt{\\lambda}x) \\right) ce^{-\\lambda \\alpha^{2} t}. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P219\" title=\"4P219\"&gt;\n                    Os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1894.svg\" alt=\"$b$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1895.svg\" alt=\"$c$\"&gt;&lt;/span&gt; poderão ser determinados ou estimados utilizando as condições iniciais e as condições de\n                    contorno. Aplicando as condições de contorno, e já descartando a função exponencial que nunca se anula, temos que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\left\\{ \\begin{array}{l}\n                    ca + cb\\sqrt{\\lambda} = 0, \\\\\n                    ca\\cos(\\sqrt{\\lambda}L) + cb{\\mathrm {sen}}(\\sqrt{\\lambda}L) - ca\\sqrt{\\lambda}{\\mathrm {sen}}(\\sqrt{\\lambda}L) +\n                    cb\\sqrt{\\lambda}\\cos(\\sqrt{\\lambda}L) = 0.\n                       \\end{array} \\right. \n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P220\" title=\"4P220\"&gt;\n                    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1896.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l}\n                    ca + cb\\sqrt{\\lambda} = 0, \\\\\n                    ca\\cos(...\n                    ...\\lambda}L) +\n                    cb\\sqrt{\\lambda}\\cos(\\sqrt{\\lambda}L) = 0.\n                    \\end{array} \\right. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P221\" title=\"4P221\"&gt;\n                    Vamos considerar que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1897.svg\" alt=\"$c \\neq 0$\"&gt;&lt;/span&gt; pois estamos interessados em uma solução não identicamente nula. Da primeira equação\n                    obtemos que &lt;!-- MATH\n                     $a = -b\\sqrt{\\lambda}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.38ex; vertical-align: -0.33ex; \" src=\"img/img1898.svg\" alt=\"$a = -b\\sqrt{\\lambda}$\"&gt;&lt;/span&gt; e, substituindo isso na segunda equação, temos\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    b(1+\\lambda) {\\mathrm {sen}}(\\sqrt{\\lambda}L) = 0.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P222\" title=\"4P222\"&gt;\n                    &lt;img style=\"height: 2.78ex; vertical-align: -0.62ex; \" src=\"img/img1899.svg\" alt=\"$\\displaystyle b(1+\\lambda) {\\mathrm {sen}}(\\sqrt{\\lambda}L) = 0. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P223\" title=\"4P223\"&gt;\n                    Mas como &lt;!-- MATH\n                     $\\lambda &gt; 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1883.svg\" alt=\"$\\lambda &gt; 0$\"&gt;&lt;/span&gt; então &lt;!-- MATH\n                     $(1+\\lambda) \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1900.svg\" alt=\"$(1+\\lambda) \\neq 0$\"&gt;&lt;/span&gt; e, por isso, também vamos impor que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1901.svg\" alt=\"$b \\neq 0$\"&gt;&lt;/span&gt;, pois caso contrário,\n                    isto é, se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1902.svg\" alt=\"$b = 0$\"&gt;&lt;/span&gt; teríamos também &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1903.svg\" alt=\"$a = 0$\"&gt;&lt;/span&gt; e a função &lt;!-- MATH\n                     $\\varphi(x)$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1904.svg\" alt=\"$\\varphi(x)$\"&gt;&lt;/span&gt; se tornaria identicamente nula, fazendo a solução\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1874.svg\" alt=\"$u(x,t)$\"&gt;&lt;/span&gt; identicamente nula. Como estamos interessados em soluções não nulas, vamos impor &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1901.svg\" alt=\"$b \\neq 0$\"&gt;&lt;/span&gt;. Desta forma,\n                    resta que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    {\\mathrm {sen}}(\\sqrt{\\lambda}L) = 0,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P224\" title=\"4P224\"&gt;\n                    &lt;img style=\"height: 2.78ex; vertical-align: -0.62ex; \" src=\"img/img1905.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(\\sqrt{\\lambda}L) = 0, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P225\" style=\"text-indent: 0 !important;\" title=\"4P225\"&gt;\n                    donde\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\sqrt{\\lambda} = \\frac{k\\pi}{L},\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P226\" title=\"4P226\"&gt;\n                    &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img1906.svg\" alt=\"$\\displaystyle \\sqrt{\\lambda} = \\frac{k\\pi}{L}, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P227\" style=\"text-indent: 0 !important;\" title=\"4P227\"&gt;\n                    para qualquer &lt;!-- MATH\n                     $k \\in \\mathbb{Z}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img48.svg\" alt=\"$k \\in \\mathbb{Z}$\"&gt;&lt;/span&gt;. Já que o sinal do argumento no seno se transmite para o coeficiente &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1894.svg\" alt=\"$b$\"&gt;&lt;/span&gt;, podemos considerar\n                    que &lt;!-- MATH\n                     $k \\in \\mathbb{N}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1907.svg\" alt=\"$k \\in \\mathbb{N}$\"&gt;&lt;/span&gt;. Sendo assim, para cada &lt;!-- MATH\n                     $k \\in \\mathbb{N}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1907.svg\" alt=\"$k \\in \\mathbb{N}$\"&gt;&lt;/span&gt;, temos uma solução\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    u_{k}(x,t) = \\left( a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) e^{-\\lambda \\alpha^{2} t},\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P228\" title=\"4P228\"&gt;\n                    &lt;img style=\"height: 5.24ex; vertical-align: -2.10ex; \" src=\"img/img1908.svg\" alt=\"$\\displaystyle u_{k}(x,t) = \\left( a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) e^{-\\lambda \\alpha^{2} t}, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P229\" style=\"text-indent: 0 !important;\" title=\"4P229\"&gt;\n                    para certos coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt;, que já incorporaram também a constante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1895.svg\" alt=\"$c$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P230\" title=\"4P230\"&gt;\n                    Pelo princípio da superposição de soluções para equações homogêneas, temos que a soma destas soluções é ainda uma\n                    solução. Segue que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    u(x,t) = \\sum_{k=0}^{\\infty} u_{k}(x,t) = \\sum_{k=0}^{\\infty} \\left( a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) e^{-\\lambda \\alpha^{2} t}.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P231\" title=\"4P231\"&gt;\n                    &lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1911.svg\" alt=\"$\\displaystyle u(x,t) = \\sum_{k=0}^{\\infty} u_{k}(x,t) = \\sum_{k=0}^{\\infty} \\le...\n                    ...L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) e^{-\\lambda \\alpha^{2} t}. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P232\" title=\"4P232\"&gt;\n                    Tudo o que precisamos agora é determinar os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt;. Aplicando a condição inicial, temos que\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P233\" title=\"4P233\"&gt;&lt;a id=\"serieFf\"&gt;&lt;/a&gt;&lt;!-- MATH\n                     \\begin{equation}\n                    f(x) = u(x,0) = \\sum_{k=0}^{\\infty} a_{k}\\cos(\\frac{k\\pi}{L}x) + b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x).\n                    \\end{equation}\n                     --&gt;\n                    &lt;table &gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.50ex; vertical-align: -2.91ex; \" src=\"img/img1912.svg\" alt=\"$\\displaystyle f(x) = u(x,0) = \\sum_{k=0}^{\\infty} a_{k}\\cos(\\frac{k\\pi}{L}x) + b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                        (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;11&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"4P234\" title=\"4P234\"&gt;\n                    Os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt; procurados, são então coeficientes que satisfazem a identidade (&lt;a href=\"#serieFf\"&gt;4.11&lt;/a&gt;).\n                    Perguntamos então quais as hipóteses sobre &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; para que a igualdade (&lt;a href=\"#serieFf\"&gt;4.11&lt;/a&gt;) se verifique para certos\n                    coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt; reais? Esta questão é respondida pelo teorema &lt;a href=\"#teoFourier\"&gt;4.7&lt;/a&gt; enunciado mais adiante.\n                    Além disso, se a igualdade (&lt;a href=\"#serieFf\"&gt;4.11&lt;/a&gt;) se verificar, como são calculados os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt;? Esta\n                    questão será comentada agora.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P235\" title=\"4P235\"&gt;\n                    Para mostrar como são calculados estes coeficientes, precisamos estudar algumas propriedades a respeito do conjunto de\n                    funções\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\mathcal{F} = \\left\\{ \\cos(\\frac{m\\pi}{L}x); \\quad m \\in \\mathbb{N}\\right\\} \\cup \\left\\{ {\\mathrm {sen}}(\\frac{n\\pi}{L}x); \\quad n \\in \\mathbb{N}\\right\\}.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P236\" title=\"4P236\"&gt;\n                    &lt;img style=\"height: 4.06ex; vertical-align: -1.55ex; \" src=\"img/img1913.svg\" alt=\"$\\displaystyle \\mathcal{F} = \\left\\{ \\cos(\\frac{m\\pi}{L}x); \\quad m \\in \\mathbb{...\n                    ...\\cup \\left\\{ {\\mathrm {sen}}(\\frac{n\\pi}{L}x); \\quad n \\in \\mathbb{N}\\right\\}. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P237\" title=\"4P237\"&gt;\n                    Consideremos o conjunto &lt;!-- MATH\n                     $\\mathcal{C} = C([-L,L]; \\mathbb{R})$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1914.svg\" alt=\"$\\mathcal{C} = C([-L,L]; \\mathbb{R})$\"&gt;&lt;/span&gt;, das funções contínuas e definidas no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1915.svg\" alt=\"$[-L,L]$\"&gt;&lt;/span&gt; com\n                    valores em &lt;!-- MATH\n                     $\\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.64ex; vertical-align: -0.10ex; \" src=\"img/img217.svg\" alt=\"$\\mathbb{R}$\"&gt;&lt;/span&gt;. Este conjunto munido da soma de funções\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    (f+g)(x) = f(x) + g(x)\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P238\" title=\"4P238\"&gt;\n                    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1916.svg\" alt=\"$\\displaystyle (f+g)(x) = f(x) + g(x) $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P239\" style=\"text-indent: 0 !important;\" title=\"4P239\"&gt;\n                    e do produto por escalar &lt;!-- MATH\n                     $a \\in \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img284.svg\" alt=\"$a \\in \\mathbb{R}$\"&gt;&lt;/span&gt;,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    (af)(x) = a f(x),\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P240\" title=\"4P240\"&gt;\n                    &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1917.svg\" alt=\"$\\displaystyle (af)(x) = a f(x), $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P241\" style=\"text-indent: 0 !important;\" title=\"4P241\"&gt;\n                    é um espaço vetorial real. O funcional bilinear &lt;!-- MATH\n                     $\\langle \\cdot, \\cdot \\rangle: \\mathcal{C} \\times \\mathcal{C} \\to \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1918.svg\" alt=\"$\\langle \\cdot, \\cdot \\rangle: \\mathcal{C} \\times \\mathcal{C} \\to \\mathbb{R}$\"&gt;&lt;/span&gt;\n                    dado por\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\langle f, g \\rangle = \\int_{-L}^{L} f(x) g(x) dx\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P242\" title=\"4P242\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1919.svg\" alt=\"$\\displaystyle \\langle f, g \\rangle = \\int_{-L}^{L} f(x) g(x) dx $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P243\" style=\"text-indent: 0 !important;\" title=\"4P243\"&gt;\n                    define um produto interno em &lt;!-- MATH\n                     $\\mathcal{C}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.63ex; vertical-align: -0.10ex; \" src=\"img/img1920.svg\" alt=\"$\\mathcal{C}$\"&gt;&lt;/span&gt;. A respeito deste produto interno vemos que o conjunto &lt;!-- MATH\n                     $\\mathcal{F}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.63ex; vertical-align: -0.10ex; \" src=\"img/img1921.svg\" alt=\"$\\mathcal{F}$\"&gt;&lt;/span&gt; é um\n                    conjunto de vetores ortogonais no espaço vetorial &lt;!-- MATH\n                     $\\mathcal{C} = C([-L,L]; \\mathbb{R})$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1914.svg\" alt=\"$\\mathcal{C} = C([-L,L]; \\mathbb{R})$\"&gt;&lt;/span&gt;. Provaremos isto em forma de teorema.\n                    &lt;/p&gt;\n                    \n                    &lt;div id=\"4Teo6\" title=\"4Teo6\" class=\"unidade\"&gt;&lt;a id=\"teoFLI\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;6&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;O conjunto de funções\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    \\mathcal{F} = \\left\\{ \\cos(\\frac{m\\pi}{L}x); \\quad m \\in \\mathbb{N}\\right\\} \\cup \\left\\{ {\\mathrm {sen}}(\\frac{n\\pi}{L}x); \\quad n \\in \\mathbb{N}\\right\\}.\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"4P244\" title=\"4P244\"&gt;\n                    &lt;img style=\"height: 4.06ex; vertical-align: -1.55ex; \" src=\"img/img1913.svg\" alt=\"$\\displaystyle \\mathcal{F} = \\left\\{ \\cos(\\frac{m\\pi}{L}x); \\quad m \\in \\mathbb{...\n                    ...\\cup \\left\\{ {\\mathrm {sen}}(\\frac{n\\pi}{L}x); \\quad n \\in \\mathbb{N}\\right\\}. $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P245\" style=\"text-indent: 0 !important;\" title=\"4P245\"&gt;&lt;i&gt;\n                    é um conjunto de funções duas a duas ortogonais do espaço vetorial &lt;!-- MATH\n                     $\\mathcal{C} = C([-L,L]; \\mathbb{R})$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1914.svg\" alt=\"$\\mathcal{C} = C([-L,L]; \\mathbb{R})$\"&gt;&lt;/span&gt;.\n                    &lt;/i&gt;&lt;/p&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;div&gt;&lt;i&gt;Prova&lt;/i&gt;.\n                    \n                    Para quaisquer &lt;!-- MATH\n                     $m,n \\in \\mathbb{N}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1922.svg\" alt=\"$m,n \\in \\mathbb{N}$\"&gt;&lt;/span&gt;, temos que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx = 0\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"4P246\" title=\"4P246\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1923.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx = 0 $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P247\" style=\"text-indent: 0 !important;\" title=\"4P247\"&gt;\n                    já que o integrando é uma função ímpar.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P248\" title=\"4P248\"&gt;\n                    Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1924.svg\" alt=\"$m \\neq n$\"&gt;&lt;/span&gt; então\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P249\" title=\"4P249\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1925.svg\" alt=\"$\\displaystyle \\int_{-L}^{L}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img1926.svg\" alt=\"$\\displaystyle \\cos(\\frac{m\\pi}{L}x) \\cos(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1927.svg\" alt=\"$\\displaystyle = \\frac{1}{2} \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x+\\frac{n\\pi}{L}x) + \\cos(\\frac{m\\pi}{L}x-\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.92ex; vertical-align: -2.33ex; \" src=\"img/img1928.svg\" alt=\"$\\displaystyle = \\frac{1}{2} \\left[ \\frac{L}{(m+n)\\pi}{\\mathrm {sen}}(\\frac{m\\pi...\n                    ...(m-n)\\pi}{\\mathrm {sen}}(\\frac{m\\pi}{L}x-\\frac{n\\pi}{L}x) \\right]_{-L}^{L} = 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P250\" style=\"text-indent: 0 !important;\" title=\"4P250\"&gt;\n                    pois a função seno se anula para argumentos múltiplos inteiros de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img9.svg\" alt=\"$\\pi $\"&gt;&lt;/span&gt;. Da mesma forma, ainda para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1924.svg\" alt=\"$m \\neq n$\"&gt;&lt;/span&gt;,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P251\" title=\"4P251\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1925.svg\" alt=\"$\\displaystyle \\int_{-L}^{L}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img1929.svg\" alt=\"$\\displaystyle {\\mathrm {sen}}(\\frac{m\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1930.svg\" alt=\"$\\displaystyle = \\frac{1}{2} \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x-\\frac{n\\pi}{L}x) - \\cos(\\frac{m\\pi}{L}x+\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.92ex; vertical-align: -2.33ex; \" src=\"img/img1931.svg\" alt=\"$\\displaystyle = \\frac{1}{2} \\left[ \\frac{L}{(m-n)\\pi}{\\mathrm {sen}}(\\frac{m\\pi...\n                    ...(m+n)\\pi}{\\mathrm {sen}}(\\frac{m\\pi}{L}x+\\frac{n\\pi}{L}x) \\right]_{-L}^{L} = 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P252\" style=\"text-indent: 0 !important;\" title=\"4P252\"&gt;\n                    e isto termina a prova.\n                    &lt;span style=\"float: right\"&gt;&lt;img style=\"height: 1.59ex; vertical-align: -0.10ex; \" src=\"img/img193.svg\" alt=\"$\\qedsymbol$\"&gt;&lt;/span&gt;\n                     &lt;/p&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P253\" title=\"4P253\"&gt;\n                    Em relação ao conjunto &lt;!-- MATH\n                     $\\mathcal{F}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.63ex; vertical-align: -0.10ex; \" src=\"img/img1921.svg\" alt=\"$\\mathcal{F}$\"&gt;&lt;/span&gt; do teorema anterior, é importante provar que o produto interno de uma destas\n                    funções com ela mesma não se anula. Faremos isto agora pois precisaremos destes resultados mais tarde. Usando as\n                    fórmulas para as integrais de seno e cosseno quadrado, obtidas na seção &lt;a href=\"aplicacoes#secinttrig\"&gt;4.2&lt;/a&gt;, temos\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P254\" title=\"4P254\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1932.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x) \\cos(\\frac{m\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1933.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} \\cos^{2}(\\frac{m\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.19ex; vertical-align: -2.14ex; \" src=\"img/img1934.svg\" alt=\"$\\displaystyle = \\frac{L}{m\\pi} \\int_{-m\\pi}^{m\\pi} \\cos^{2}u du$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.57ex; vertical-align: -2.33ex; \" src=\"img/img1935.svg\" alt=\"$\\displaystyle = \\frac{L}{m\\pi} \\left[ \\frac{1}{2}u + \\frac{1}{4} {\\mathrm {sen}}(2u) \\right]_{-m\\pi}^{m\\pi} = L,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P255\" style=\"text-indent: 0 !important;\" title=\"4P255\"&gt;\n                    desde que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1936.svg\" alt=\"$m \\neq 0$\"&gt;&lt;/span&gt;. Se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1937.svg\" alt=\"$m = 0$\"&gt;&lt;/span&gt; então claramente\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x) \\cos(\\frac{m\\pi}{L}x) dx = \\int_{-L}^{L}  dx  = 2L.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P256\" title=\"4P256\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1938.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} \\cos(\\frac{m\\pi}{L}x) \\cos(\\frac{m\\pi}{L}x) dx = \\int_{-L}^{L} dx = 2L. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P257\" title=\"4P257\"&gt;\n                    Da mesma forma temos que\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P258\" title=\"4P258\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1939.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{m\\pi}{L}x) {\\mathrm {sen}}(\\frac{m\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1940.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} {\\mathrm {sen}}^{2}(\\frac{m\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.19ex; vertical-align: -2.14ex; \" src=\"img/img1941.svg\" alt=\"$\\displaystyle = \\frac{L}{m\\pi} \\int_{-m\\pi}^{m\\pi} {\\mathrm {sen}}^{2}u du$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.57ex; vertical-align: -2.33ex; \" src=\"img/img1942.svg\" alt=\"$\\displaystyle = \\frac{L}{m\\pi} \\left[ \\frac{1}{2}u - \\frac{1}{4} {\\mathrm {sen}}(2u) \\right]_{-m\\pi}^{m\\pi} = L,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P259\" style=\"text-indent: 0 !important;\" title=\"4P259\"&gt;\n                    desde que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1936.svg\" alt=\"$m \\neq 0$\"&gt;&lt;/span&gt;. O caso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1937.svg\" alt=\"$m = 0$\"&gt;&lt;/span&gt; fica\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{m\\pi}{L}x) {\\mathrm {sen}}(\\frac{m\\pi}{L}x) dx = \\int_{-L}^{L} 0 dx  = 0.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P260\" title=\"4P260\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1943.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{m\\pi}{L}x) {\\mathrm {sen}}(\\frac{m\\pi}{L}x) dx = \\int_{-L}^{L} 0 dx = 0. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P261\" title=\"4P261\"&gt;\n                    Vamos agora obter as expressões para os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt;, admitindo que a função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; possa ser escrita na\n                    forma da série (&lt;a href=\"#serieFf\"&gt;4.11&lt;/a&gt;), ou ainda,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P262\" title=\"4P262\"&gt;&lt;a id=\"serieFfa0\"&gt;&lt;/a&gt;&lt;!-- MATH\n                     \\begin{equation}\n                    f(x) = a_{0} + \\sum_{k=1}^{\\infty} \\left( a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right).\n                    \\end{equation}\n                     --&gt;\n                    &lt;table &gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1944.svg\" alt=\"$\\displaystyle f(x) = a_{0} + \\sum_{k=1}^{\\infty} \\left( a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                        (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;12&lt;/span&gt;)  &lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"4P263\" title=\"4P263\"&gt;\n                    Primeiramente vamos obter &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1122.svg\" alt=\"$a_{0}$\"&gt;&lt;/span&gt;. Integrando (&lt;a href=\"#serieFfa0\"&gt;4.12&lt;/a&gt;) em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt;, de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.27ex; \" src=\"img/img1945.svg\" alt=\"$-L$\"&gt;&lt;/span&gt; a &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1841.svg\" alt=\"$L$\"&gt;&lt;/span&gt;, obtemos\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\int_{-L}^{L} f(x) dx = \\int_{-L}^{L} a_{0} + \\sum_{k=1}^{\\infty} \\left( a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) dx.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P264\" title=\"4P264\"&gt;\n                    &lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1946.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} f(x) dx = \\int_{-L}^{L} a_{0} + \\sum_{k=1}^{\\infty}...\n                    ... a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) dx. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P265\" title=\"4P265\"&gt;\n                    Como o somatório acima converge uniformemente então podemos integrar termo a termo. Temos assim,\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P266\" title=\"4P266\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1947.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} f(x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1948.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} a_{0} dx + \\sum_{k=1}^{\\infty} \\left( a_{k} \\int_...\n                    ...k\\pi}{L}x) dx + b_{k} \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{k\\pi}{L}x) dx \\right)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1949.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} a_{0} dx = 2La_{0},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P267\" style=\"text-indent: 0 !important;\" title=\"4P267\"&gt;\n                    já que as integrais trigonométricas do somatório se anulam. Temos portanto que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    a_{0} = \\frac{1}{2L} \\int_{-L}^{L} f(x) dx.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P268\" title=\"4P268\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1950.svg\" alt=\"$\\displaystyle a_{0} = \\frac{1}{2L} \\int_{-L}^{L} f(x) dx. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P269\" title=\"4P269\"&gt;\n                    Para obter cada um dos demais termos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\"&gt;&lt;/span&gt;, para &lt;!-- MATH\n                     $n=1,2,3,\\dots$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1951.svg\" alt=\"$n=1,2,3,\\dots$\"&gt;&lt;/span&gt;, tomamos o produto interno de (&lt;a href=\"#serieFfa0\"&gt;4.12&lt;/a&gt;) com\n                    a respectiva função &lt;!-- MATH\n                     $\\cos(\\frac{n\\pi}{L}x)$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1952.svg\" alt=\"$\\cos(\\frac{n\\pi}{L}x)$\"&gt;&lt;/span&gt;. Temos então\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P270\" title=\"4P270\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1953.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} f(x)\\cos(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.67ex; vertical-align: -2.90ex; \" src=\"img/img1954.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} \\left( a_{0} + \\sum_{k=1}^{\\infty} a_{k}\\cos(\\fra...\n                    ...i}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right) \\cos(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1955.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} a_{0}\\cos(\\frac{n\\pi}{L}x) dx + \\sum_{k=1}^{\\infty} a_{k} \\int_{-L}^{L} \\cos(\\frac{k\\pi}{L}x) \\cos(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1956.svg\" alt=\"$\\displaystyle \\qquad \\qquad + \\sum_{k=1}^{\\infty} b_{k} \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\cos(\\frac{n\\pi}{L}x) dx.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"4P271\" title=\"4P271\"&gt;\n                    De acordo com os resultados do teorema &lt;a href=\"#teoFLI\"&gt;4.6&lt;/a&gt;, as integrais do último membro se anulam todas, exceto a integral\n                    em cossenos quando &lt;!-- MATH\n                     $k = n \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1957.svg\" alt=\"$k = n \\neq 0$\"&gt;&lt;/span&gt;. Segue que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\int_{-L}^{L} f(x)\\cos(\\frac{n\\pi}{L}x) dx = a_{n} \\int_{-L}^{L} \\cos(\\frac{n\\pi}{L}x) \\cos(\\frac{n\\pi}{L}x) dx = a_{n} L,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P272\" title=\"4P272\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1958.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} f(x)\\cos(\\frac{n\\pi}{L}x) dx = a_{n} \\int_{-L}^{L} \\cos(\\frac{n\\pi}{L}x) \\cos(\\frac{n\\pi}{L}x) dx = a_{n} L, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P273\" style=\"text-indent: 0 !important;\" title=\"4P273\"&gt;\n                    e, portanto,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    a_{n} = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos(\\frac{n\\pi}{L}x) dx,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P274\" title=\"4P274\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1959.svg\" alt=\"$\\displaystyle a_{n} = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos(\\frac{n\\pi}{L}x) dx, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P275\" style=\"text-indent: 0 !important;\" title=\"4P275\"&gt;\n                    para todo &lt;!-- MATH\n                     $n = 1,2,3, \\dots$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1951.svg\" alt=\"$n=1,2,3,\\dots$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P276\" title=\"4P276\"&gt;\n                    Analogamente obtemos cada um dos coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.96ex; vertical-align: -0.41ex; \" src=\"img/img1960.svg\" alt=\"$b_{n}$\"&gt;&lt;/span&gt;, tomando o produto interno de (&lt;a href=\"#serieFfa0\"&gt;4.12&lt;/a&gt;) com a respectiva\n                    função &lt;!-- MATH\n                     ${\\mathrm {sen}}(\\frac{n\\pi}{L}x)$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.50ex; vertical-align: -0.83ex; \" src=\"img/img1961.svg\" alt=\"${\\mathrm {sen}}(\\frac{n\\pi}{L}x)$\"&gt;&lt;/span&gt;. Temos então que\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P277\" title=\"4P277\"&gt;&lt;table class=\"equation\"&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1962.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} f(x){\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1963.svg\" alt=\"$\\displaystyle = \\int_{-L}^{L} a_{0}{\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx + \\sum_{...\n                    ...} a_{k} \\int_{-L}^{L} \\cos(\\frac{k\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;tr&gt;\n                    &lt;td&gt;&nbsp;&lt;/td&gt;\n                    &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1964.svg\" alt=\"$\\displaystyle \\qquad \\qquad + \\sum_{k=1}^{\\infty} b_{k} \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{k\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                    &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    \n                    \n                    &lt;p class=\" unidade\" id=\"4P278\" title=\"4P278\"&gt;\n                    De acordo com os resultados obtidos no teorema &lt;a href=\"#teoFLI\"&gt;4.6&lt;/a&gt;, as integrais do último membro se anulam todas, exceto a\n                    integral em senos quando &lt;!-- MATH\n                     $k = n \\neq 0$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.51ex; \" src=\"img/img1957.svg\" alt=\"$k = n \\neq 0$\"&gt;&lt;/span&gt;. Temos assim,\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\int_{-L}^{L} f(x){\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx = b_{n} \\int_{-L}^{L} {\\mathrm {sen}}(\\frac{n\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx = b_{n}L.\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P279\" title=\"4P279\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1965.svg\" alt=\"$\\displaystyle \\int_{-L}^{L} f(x){\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx = b_{n} \\in...\n                    ...{\\mathrm {sen}}(\\frac{n\\pi}{L}x) {\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx = b_{n}L. $\"&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P280\" title=\"4P280\"&gt;\n                    Segue que\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    b_{n} = \\frac{1}{L} \\int_{-L}^{L} f(x){\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P281\" title=\"4P281\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1966.svg\" alt=\"$\\displaystyle b_{n} = \\frac{1}{L} \\int_{-L}^{L} f(x){\\mathrm {sen}}(\\frac{n\\pi}{L}x) dx, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P282\" style=\"text-indent: 0 !important;\" title=\"4P282\"&gt;\n                    para &lt;!-- MATH\n                     $n = 1,2,3, \\dots$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1951.svg\" alt=\"$n=1,2,3,\\dots$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P283\" title=\"4P283\"&gt;\n                    Desta forma temos que\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P284\" title=\"4P284\"&gt;&lt;a id=\"serieF\"&gt;&lt;/a&gt;&lt;!-- MATH\n                     \\begin{equation}\n                    f(x) = \\frac{a_{0}}{2} + \\sum_{k=1}^{\\infty} a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x)\n                    \\end{equation}\n                     --&gt;\n                    &lt;table&gt;\n                    &lt;tbody&gt;&lt;tr&gt;\n                    &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1967.svg\" alt=\"$\\displaystyle f(x) = \\frac{a_{0}}{2} + \\sum_{k=1}^{\\infty} a_{k}\\cos(\\frac{k\\pi}{L}x) +b_{k}{\\mathrm {sen}}(\\frac{k\\pi}{L}x)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                    &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                        (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;13&lt;/span&gt;)  &lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                    &lt;p class=\" unidade\" id=\"4P285\" style=\"text-indent: 0 !important;\" title=\"4P285\"&gt;\n                    para\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    a_{k} = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos(\\frac{k\\pi}{L} x) dx \\qquad (k \\geq 0),\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P286\" title=\"4P286\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1968.svg\" alt=\"$\\displaystyle a_{k} = \\frac{1}{L} \\int_{-L}^{L} f(x) \\cos(\\frac{k\\pi}{L} x) dx \\qquad (k \\geq 0), $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P287\" style=\"text-indent: 0 !important;\" title=\"4P287\"&gt;\n                    e\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    b_{k} = \\frac{1}{L} \\int_{-L}^{L} f(x) {\\mathrm {sen}}(\\frac{k\\pi}{L} x) dx \\qquad (k &gt; 0),\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P288\" title=\"4P288\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1969.svg\" alt=\"$\\displaystyle b_{k} = \\frac{1}{L} \\int_{-L}^{L} f(x) {\\mathrm {sen}}(\\frac{k\\pi}{L} x) dx \\qquad (k &gt; 0), $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P289\" style=\"text-indent: 0 !important;\" title=\"4P289\"&gt;\n                    desde que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; admita representação na forma da série (&lt;a href=\"#serieFf\"&gt;4.11&lt;/a&gt;). O ajuste &lt;!-- MATH\n                     $\\frac{1}{2}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img1724.svg\" alt=\"$\\frac{1}{2}$\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1122.svg\" alt=\"$a_{0}$\"&gt;&lt;/span&gt; é só para\n                    padronizar a expressão dos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; para todo &lt;!-- MATH\n                     $k \\in \\mathbb{N}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img1907.svg\" alt=\"$k \\in \\mathbb{N}$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P290\" title=\"4P290\"&gt;\n                    Os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt; obtidos pelas expressões acima são chamados de coeficientes de Fourier da função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;.\n                    A série em (&lt;a href=\"#serieF\"&gt;4.13&lt;/a&gt;) é chamada de série de Fourier de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;. Muito cuidado neste momento. A série do lado direito de\n                    (&lt;a href=\"#serieF\"&gt;4.13&lt;/a&gt;) simplesmente é definida como sendo a série de Fourier da função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;. Não falamos nada a respeito da\n                    série de Fourier de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; convergir para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;, aliás, admitimos esta convergência para calcular os coeficientes. Isto\n                    significa que garantir a igualdade (&lt;a href=\"#serieF\"&gt;4.13&lt;/a&gt;) é um pouco mais complicado do que parece.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P291\" title=\"4P291\"&gt;\n                    Apresentamos agora um teorema que garante esta convergência. A demonstração pode ser encontrada em [&lt;a href=\"/trigonometria-hiperbolica/referencias#Iorio\"&gt;3&lt;/a&gt;, Iorio]\n                    ou [&lt;a href=\"/trigonometria-hiperbolica/referencias#Guidorizzi\"&gt;2&lt;/a&gt;, Guidorizzi].\n                    &lt;/p&gt;\n                    \n                    &lt;div id=\"4Teo7\" title=\"4Teo7\" class=\"unidade\"&gt;&lt;a id=\"teoFourier\"&gt;&lt;b&gt;Teorema  &lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;7&lt;/span&gt;&lt;/b&gt;&lt;/a&gt; &nbsp; \n                    &lt;i&gt;Seja &lt;!-- MATH\n                     $f:[-L,L] \\to \\mathbb{R}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1970.svg\" alt=\"$f:[-L,L] \\to \\mathbb{R}$\"&gt;&lt;/span&gt; uma função contínua, com derivada segunda contínua por partes e tal que &lt;!-- MATH\n                     $f(-L) = f(L)$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1971.svg\" alt=\"$f(-L) = f(L)$\"&gt;&lt;/span&gt;. Então a\n                    série de Fourier de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; converge uniformemente para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1915.svg\" alt=\"$[-L,L]$\"&gt;&lt;/span&gt;. Isto é,\n                    &lt;/i&gt;&lt;!-- MATH\n                     \\begin{displaymath}\n                    f(x) = \\frac{a_{0}}{2} + \\sum_{k=1}^{\\infty} \\left[ a_{k} \\cos(\\frac{k\\pi}{L}x) + b_{k} {\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right],\n                    \\end{displaymath}\n                     --&gt;\n                    \n                    &lt;div class=\"mathdisplay unidade\" id=\"4P292\" title=\"4P292\"&gt;\n                    &lt;img style=\"height: 6.49ex; vertical-align: -2.90ex; \" src=\"img/img1972.svg\" alt=\"$\\displaystyle f(x) = \\frac{a_{0}}{2} + \\sum_{k=1}^{\\infty} \\left[ a_{k} \\cos(\\frac{k\\pi}{L}x) + b_{k} {\\mathrm {sen}}(\\frac{k\\pi}{L}x) \\right], $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P293\" style=\"text-indent: 0 !important;\" title=\"4P293\"&gt;&lt;i&gt;\n                    para todo &lt;!-- MATH\n                     $x \\in [-L,L]$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1973.svg\" alt=\"$x \\in [-L,L]$\"&gt;&lt;/span&gt;.\n                    &lt;/i&gt;&lt;/p&gt;&lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P294\" title=\"4P294\"&gt;\n                    A hipótese de que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; seja contínua é bastante forte. Na verdade, esta hipótese pode ser reduzida para uma hipótese de\n                    continuidade por partes. A hipótese de continuidade por partes da derivada segunda já garante isto. Neste caso, em cada\n                    ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img69.svg\" alt=\"$x$\"&gt;&lt;/span&gt; de continuidade de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; a série de Fourier de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; converge para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; e nos pontos de descontinuidade\n                    (descontinuidade tipo salto) a série de Fourier converge para o ponto médio dos limites laterais de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;. Este resultado\n                    pode ser encontrado também em [&lt;a href=\"/trigonometria-hiperbolica/referencias#Iorio\"&gt;3&lt;/a&gt;, Iorio].\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P295\" title=\"4P295\"&gt;\n                    Podemos também colocar o intervalo de interesse como sendo um intervalo da forma &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1974.svg\" alt=\"$[0,L]$\"&gt;&lt;/span&gt;. Isto não é problema pois dada\n                    uma função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; definida no intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1974.svg\" alt=\"$[0,L]$\"&gt;&lt;/span&gt; podemos construir uma extensão &lt;!-- MATH\n                     $\\overline{f}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.07ex; vertical-align: -0.10ex; \" src=\"img/img1975.svg\" alt=\"$\\overline{f}$\"&gt;&lt;/span&gt; de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; a todo intervalo\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1915.svg\" alt=\"$[-L,L]$\"&gt;&lt;/span&gt; por\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    \\overline{f}(x)\n                     = \\left\\{ \\begin{array}{lcl} 0 & \\text{se} & -L \\leq x &lt; 0 \\\\f(x) & \\text{se} & 0 \\leq x \\leq L, \\end{array} \\right. \n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P296\" title=\"4P296\"&gt;\n                    &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img1976.svg\" alt=\"$\\displaystyle \\overline{f}(x)\n                    = \\left\\{ \\begin{array}{lcl} 0 &amp; \\text{se} &amp; -L \\leq x &lt; 0 \\\\ f(x) &amp; \\text{se} &amp; 0 \\leq x \\leq L, \\end{array} \\right. $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P297\" style=\"text-indent: 0 !important;\" title=\"4P297\"&gt;\n                    e os coeficientes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img1909.svg\" alt=\"$a_{k}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img1910.svg\" alt=\"$b_{k}$\"&gt;&lt;/span&gt; de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1977.svg\" alt=\"$\\bar{f}$\"&gt;&lt;/span&gt; ficam também reduzidos a\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    a_{k} = \\frac{1}{L} \\int_{-L}^{L} \\overline{f}(x) \\cos(\\frac{k\\pi}{L} x) dx = \\frac{1}{L} \\int_{0}^{L} f(x) \\cos(\\frac{k\\pi}{L} x) dx,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P298\" title=\"4P298\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1978.svg\" alt=\"$\\displaystyle a_{k} = \\frac{1}{L} \\int_{-L}^{L} \\overline{f}(x) \\cos(\\frac{k\\pi}{L} x) dx = \\frac{1}{L} \\int_{0}^{L} f(x) \\cos(\\frac{k\\pi}{L} x) dx, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P299\" style=\"text-indent: 0 !important;\" title=\"4P299\"&gt;\n                    e\n                    &lt;!-- MATH\n                     \\begin{displaymath}\n                    b_{k} = \\frac{1}{L} \\int_{-L}^{L} \\overline{f}(x) {\\mathrm {sen}}(\\frac{k\\pi}{L} x) dx = \\frac{1}{L} \\int_{0}^{L} f(x) {\\mathrm {sen}}(\\frac{k\\pi}{L} x) dx,\n                    \\end{displaymath}\n                     --&gt;\n                    &lt;/p&gt;\n                    &lt;div class=\"mathdisplay unidade\" id=\"4P300\" title=\"4P300\"&gt;\n                    &lt;img style=\"height: 5.55ex; vertical-align: -2.14ex; \" src=\"img/img1979.svg\" alt=\"$\\displaystyle b_{k} = \\frac{1}{L} \\int_{-L}^{L} \\overline{f}(x) {\\mathrm {sen}}...\n                    ...L} x) dx = \\frac{1}{L} \\int_{0}^{L} f(x) {\\mathrm {sen}}(\\frac{k\\pi}{L} x) dx, $\"&gt;\n                    &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P301\" style=\"text-indent: 0 !important;\" title=\"4P301\"&gt;\n                    que são na verdade coeficientes para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; restritos ao intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1974.svg\" alt=\"$[0,L]$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P302\" title=\"4P302\"&gt;\n                    Embora a série de Taylor desempenha um papel fundamental para a matemática aplicada, a série de Fourier apresenta\n                    propriedades que a tornam mais adequada para certas aplicações. Dentre estas propriedades, um fato relevante é que os\n                    coeficientes da série de Taylor são os termos &lt;!-- MATH\n                     $\\frac{f^{(n)}(x_{0})}{n!}$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.36ex; vertical-align: -0.83ex; \" src=\"img/img1980.svg\" alt=\"$\\frac{f^{(n)}(x_{0})}{n!}$\"&gt;&lt;/span&gt; e então a função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; deve possuir derivadas\n                    de ordem &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1118.svg\" alt=\"$n$\"&gt;&lt;/span&gt; contínuas no ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1225.svg\" alt=\"$x_{0}$\"&gt;&lt;/span&gt;. A série de Fourier não exige tanto. O teorema que apresentamos exige apenas\n                    derivada segunda contínua por partes.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P303\" title=\"4P303\"&gt;\n                    Além disso, a série de Taylor é uma série com boa aproximação para a função nas proximidades do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1225.svg\" alt=\"$x_{0}$\"&gt;&lt;/span&gt;. Quanto\n                    mais afastado do ponto &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1225.svg\" alt=\"$x_{0}$\"&gt;&lt;/span&gt; mais coeficientes serão necessários para uma aproximação satisfatória. Já a série de\n                    Fourier tem comportamento global. Isto significa que não é necessário aumentar o número de coeficientes quando se muda\n                    o ponto de interesse do intervalo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1915.svg\" alt=\"$[-L,L]$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P304\" title=\"4P304\"&gt;\n                    Uma aplicação prática do uso da série de Fourier é o armazenamento e transmissão de imagens. Consideremos que uma\n                    fotografia seja tirada em um telefone celular com câmera. A foto é composta de pontos coloridos. Supondo que a\n                    resolução da foto seja de &lt;!-- MATH\n                     $640 \\times 480$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1981.svg\" alt=\"$640 \\times 480$\"&gt;&lt;/span&gt;, então a foto é um retângulo de 640 pontos de largura por 480 pontos de\n                    altura. São portanto 307.200 pontos coloridos, dispostos em 640 colunas e 480 linhas.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P305\" title=\"4P305\"&gt;\n                    Para gravar esta foto, no sistema VGA, são armazenados 3 bytes de informações para cada um destes pontos. Estes bytes\n                    correspondem às intensidades de vermelho, verde e azul de cada ponto. São então 921.600 bytes que devem ser gravados,\n                    isto sem contar outras informações, conhecidas como o cabeçalho da imagem. Cada byte armazena como informação um número\n                    inteiro de 0 a 255.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P306\" title=\"4P306\"&gt;\n                    Vamos agora ver como a série de Fourier pode ajudar a economizar espaço para gravar esta figura.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P307\" title=\"4P307\"&gt;\n                    Se olharmos para cada uma das 480 linhas que compõem a figura temos que cada linha possui 640 pontos. São 640 bytes\n                    armazenando as intensidades de vermelho, 640 armazenando as intensidades de verde e 640 as intensidades de azul, num\n                    total de 1.920 bytes para cada uma das 480 linhas da figura. Se traçarmos um gráfico destes 640 valores para a\n                    intensidade de vermelho, podemos construir uma função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img299.svg\" alt=\"$f(x)$\"&gt;&lt;/span&gt; que representa estas intensidades.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P308\" title=\"4P308\"&gt;\n                    A função será portanto uma função constante em cada um dos 640 subintervalos no qual foi dividido o intervalo &lt;!-- MATH\n                     $[0,L] =\n                    [0,640]$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1982.svg\" alt=\"$[0,L] =\n                    [0,640]$\"&gt;&lt;/span&gt;.\n                    &lt;/p&gt;\n                    &lt;div class=\"CENTER\"&gt;&lt;a id=\"7016\"&gt;&lt;/a&gt;\n                    &lt;table&gt;\n                    &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 4.3:&lt;/strong&gt;\n                     Gráfico de uma cor de uma linha da figura. &lt;/caption&gt;\n                    &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                    &lt;div class=\"CENTER\"&gt;\n                    &lt;img src=\"img/SFcor.png\" alt=\"Image SFcor\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                    &lt;/tbody&gt;&lt;/table&gt;\n                    &lt;/div&gt;\n                    \n                    &lt;p class=\" unidade\" id=\"4P309\" title=\"4P309\"&gt;\n                    A função pode ainda ser reescalonada para que os valores estejam, digamos de 0 a 1, ao invéz de 0 a 255. O\n                    reescalonamento é para que os saltos da função sejam pequenos, diminuindo a oscilação da série de Fourier na passagem\n                    de um segmento a outro.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P310\" title=\"4P310\"&gt;\n                    Feito isto, montamos a série de Fourier desta função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;. A série de Fourier de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt; dará uma boa aproximação para a\n                    função &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img277.svg\" alt=\"$f$\"&gt;&lt;/span&gt;. Tudo o que precisamos armazenar agora são os coeficientes de Fourier desta série. Exemplificaremos o\n                    processo considerando 21 coeficientes, sendo eles &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1122.svg\" alt=\"$a_{0}$\"&gt;&lt;/span&gt;, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.48ex; vertical-align: -0.41ex; \" src=\"img/img1117.svg\" alt=\"$a_{n}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.96ex; vertical-align: -0.41ex; \" src=\"img/img1960.svg\" alt=\"$b_{n}$\"&gt;&lt;/span&gt; para &lt;!-- MATH\n                     $n = 1, 2, \\dots, 10$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img1983.svg\" alt=\"$n = 1, 2, \\dots, 10$\"&gt;&lt;/span&gt;. Como estes\n                    coeficientes são números reais, então o armazenamento de cada um destes números ocupa 6 bytes, totalizando 126 bytes\n                    para os 21 coeficientes de Fourier.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P311\" title=\"4P311\"&gt;\n                    Desta forma, ao invés de gastar 640 bytes para armazenar as intensidades de vermelho, podemos usar apenas 126 bytes.\n                    Repetindo este processo para as intensidades de verde e de azul, temos o armazenamento de 378 bytes para cada linha da\n                    figura, no lugar dos 1.920 bytes tradicionais.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P312\" title=\"4P312\"&gt;\n                    Aplicando isto a todas as 480 linhas da figura, temos um gasto de 181.440 bytes ao invés dos 921.600 tradicionais. Uma\n                    compactação de mais de 80%. Levemos ainda em conta que podem ser armazenados mais do que os 21 coeficientes que\n                    citamos, ou menos. Gravando mais coeficientes implicará em uma maior qualidade da imagem, porém, menos economia de\n                    espaço e menos coeficientes implicarão menor qualidade, porém, maior economia de espaço. Pode-se ainda trabalhar com\n                    apenas os coeficientes da função seno ou somente com os coeficientes da função cosseno. O formato de imagens conhecido\n                    como JPG ou JPEG é um sistema de compressão e armazenamento de imagens baseado em uma série de cossenos.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P313\" title=\"4P313\"&gt;\n                    O espaço necessário para gravar uma figura já não é um grande problema. Os atuais discos rígidos e os cartões de\n                    memória para celulares e câmeras digitais já possuem uma capacidade de armazenamento bem expressiva, o que poderia até\n                    dispensar uma compactação da imagem. Mas a transmissão das imagens é ainda um problema.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P314\" title=\"4P314\"&gt;\n                    Celulares com tecnologia GSM transmitem dados a uma taxa máxima de 9.600bps, isto é, 9.600 bits por segundo e isto\n                    significa 1.200 bytes por segundo (1byte = 8bits). A tecnologia GPRS possui na prática uma taxa de transmissão de dados\n                    de até 40.200bps (em teoria até 171.200bps). A tecnologia EDGE transmite na prática até 384.000bps (em teoria até\n                    473.600bps). A tecnologia 3G transmite até 7Mbps (Mega bits por segundo), isto é, 7.340.032 bits por segundo, ou\n                    917.504 bytes por segundo.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P315\" title=\"4P315\"&gt;\n                    Uma imagem com 921.600 bytes necessita de 768 segundos, ou 12 minutos e 48 segundos para a transmissão via GSM. Já uma\n                    imagem de 181.440 bytes, compactada por série de Fourier, necessita de 152 segundos, ou 2 minutos e 32 segundos para a\n                    transmissão. Isto conseguindo a taxa máxima de transmissão. Independentemente da tecnologia utilizada ou da taxa de\n                    transmissão atingida, 921.600 bytes sempre necessitarão 5 vezes mais tempo para serem transmitidos do que 181.440\n                    bytes.\n                    &lt;/p&gt;\n                    &lt;p class=\" unidade\" id=\"4P316\" title=\"4P316\"&gt;\n                    A evolução das tecnologias de celulares ajuda na redução do tempo de transmissão. Na contramão desta evolução, a\n                    resolução das imagens também evolui. Resoluções maiores como &lt;!-- MATH\n                     $800 \\times 600$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1984.svg\" alt=\"$800 \\times 600$\"&gt;&lt;/span&gt; ou &lt;!-- MATH\n                     $1.280 \\times 960$\n                     --&gt;\n                    &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img1985.svg\" alt=\"$1.280 \\times 960$\"&gt;&lt;/span&gt;, significam que as\n                    imagens possuem mais pontos e, consequentemente, exigem mais espaço para a gravação e mais tempo para a transmissão. A\n                    compactação continua sendo importante neste processo pois diminui o tamanho e o tempo de transmissão das imagens,\n                    quaisquer que sejam as tecnologias utilizadas para armazenamento e transmissão de dados.\n                    &lt;/p&gt;",
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    "section": "4.5 Lançamento vertical e queda livre com resistência",
    "text": "4.5 Lançamento vertical e queda livre com resistência\n\n                        &lt;p class=\" unidade\" id=\"4P317\" title=\"4P317\"&gt;\n                        Nesta seção vamos estudar a velocidade de um objeto lançado verticalmente para cima, em determinado momento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.46ex; \" src=\"img/img1986.svg\" alt=\"$t_{0}$\"&gt;&lt;/span&gt;, a\n                        uma certa velocidade inicial &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.46ex; \" src=\"img/img1987.svg\" alt=\"$v_{0}$\"&gt;&lt;/span&gt;. Quando este objeto atingir a altura máxima, ele começa a cair. Vamos também\n                        estudar esta velocidade de queda.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P318\" title=\"4P318\"&gt;\n                        Consideremos um sistema coordenado com apenas um eixo vertical, apontando para cima, cuja origem é o nível do solo.\n                        Designemos por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt; o tempo, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1988.svg\" alt=\"$y(t)$\"&gt;&lt;/span&gt; a posição (altura em relação ao solo) do objeto e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1989.svg\" alt=\"$v(t)$\"&gt;&lt;/span&gt; a velocidade do objeto no\n                        instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P319\" title=\"4P319\"&gt;\n                        Supondo que este objeto seja então lançado para cima, ele está sujeito à ação da gravidade e também a uma força de\n                        atrito com o ar. Em geral, forças de atrito são consideradas como sendo proporcionais a uma potência da velocidade.\n                        Esta potência varia de acordo com a própria velocidade. Para problemas desta natureza a potência considerada é 2.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P320\" title=\"4P320\"&gt;\n                        A ação da gravidade resume-se na força peso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img1990.svg\" alt=\"$p = -mg$\"&gt;&lt;/span&gt;, sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; a massa do objeto e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img285.svg\" alt=\"$g$\"&gt;&lt;/span&gt; a aceleração da gravidade. A\n                        força de atrito &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img40.svg\" alt=\"$r$\"&gt;&lt;/span&gt; com o ar, é considerada como &lt;!-- MATH\n                         $r = -kv^{2}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img1991.svg\" alt=\"$r = -kv^{2}$\"&gt;&lt;/span&gt;, sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img635.svg\" alt=\"$k&gt;0$\"&gt;&lt;/span&gt; a constante de proporcionalidade que é\n                        dependente de alguns fatores como densidade do ar e a área da secção transversal frontal do objeto. O sinal negativo\n                        das duas forças é decorrente do fato que são contrárias ao referencial, isto é, ambas apontam para baixo.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P321\" title=\"4P321\"&gt;\n                        Para um corpo qualquer, a constante de proporcionalidade é dada por &lt;!-- MATH\n                         $k = \\frac{\\rho A \\delta}{2}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.93ex; vertical-align: -0.83ex; \" src=\"img/img1992.svg\" alt=\"$k = \\frac{\\rho A \\delta}{2}$\"&gt;&lt;/span&gt;, sendo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img1993.svg\" alt=\"$\\rho$\"&gt;&lt;/span&gt; é a\n                        densidade do ar, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\"&gt;&lt;/span&gt; é a área da secção transversal frontal exposta ao ar e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1994.svg\" alt=\"$\\delta$\"&gt;&lt;/span&gt; é um coeficiente que depende da\n                        forma do objeto. Embora a densidade do ar varie com a altura, para cálculos aproximados em baixa altitude, pode ser\n                        utilizado o valor ao nível do mar que é de &lt;!-- MATH\n                         $1,29 Kg/m^{3}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img1995.svg\" alt=\"$1,29 Kg/m^{3}$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P322\" title=\"4P322\"&gt;\n                        De acordo com a segunda lei de Newton, temos que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1996.svg\" alt=\"$F = ma$\"&gt;&lt;/span&gt;, sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1997.svg\" alt=\"$F$\"&gt;&lt;/span&gt; a força resultante do sistema, que é a soma das\n                        duas forças consideradas. Temos assim,\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        ma = F = p + r = -mg - kv^{2}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P323\" title=\"4P323\"&gt;\n                        &lt;img style=\"height: 2.53ex; vertical-align: -0.50ex; \" src=\"img/img1998.svg\" alt=\"$\\displaystyle ma = F = p + r = -mg - kv^{2}. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P324\" title=\"4P324\"&gt;\n                        A aceleração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img295.svg\" alt=\"$a$\"&gt;&lt;/span&gt; por sua vez é igual à derivada da velocidade em função do tempo e então temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        m v' = ma = -mg - kv^{2}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P325\" title=\"4P325\"&gt;\n                        &lt;img style=\"height: 2.53ex; vertical-align: -0.50ex; \" src=\"img/img1999.svg\" alt=\"$\\displaystyle m v' = ma = -mg - kv^{2}. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P326\" title=\"4P326\"&gt;\n                        Esta é uma equação diferencial não linear, sujeita a uma condição inicial &lt;!-- MATH\n                         $v(t_{0}) = v_{0}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2000.svg\" alt=\"$v(t_{0}) = v_{0}$\"&gt;&lt;/span&gt;. Resolver esta equação nos\n                        fornece &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1989.svg\" alt=\"$v(t)$\"&gt;&lt;/span&gt; que é a velocidade do objeto em função do tempo. Vamos obter esta solução. Reescrevemos a equação\n                        diferencial na forma\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        v' = -g(1 + \\frac{k}{mg}v^{2}),\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P327\" title=\"4P327\"&gt;\n                        &lt;img style=\"height: 4.91ex; vertical-align: -1.95ex; \" src=\"img/img2001.svg\" alt=\"$\\displaystyle v' = -g(1 + \\frac{k}{mg}v^{2}), $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P328\" style=\"text-indent: 0 !important;\" title=\"4P328\"&gt;\n                        ou, ainda,\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\frac{v'}{(1 + \\frac{k}{mg}v^{2})} = -g.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P329\" title=\"4P329\"&gt;\n                        &lt;img style=\"height: 5.91ex; vertical-align: -2.78ex; \" src=\"img/img2002.svg\" alt=\"$\\displaystyle \\frac{v'}{(1 + \\frac{k}{mg}v^{2})} = -g. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P330\" title=\"4P330\"&gt;\n                        Para simplificar, chamemos &lt;!-- MATH\n                         $\\lambda^{2} = \\frac{k}{mg}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.10ex; vertical-align: -1.12ex; \" src=\"img/img2003.svg\" alt=\"$\\lambda^{2} = \\frac{k}{mg}$\"&gt;&lt;/span&gt; e integrando a equação anterior em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\int \\frac{v'}{(1 + \\lambda^{2}v^{2})} dt = -gt + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P331\" title=\"4P331\"&gt;\n                        &lt;img style=\"height: 5.20ex; vertical-align: -2.07ex; \" src=\"img/img2004.svg\" alt=\"$\\displaystyle \\int \\frac{v'}{(1 + \\lambda^{2}v^{2})} dt = -gt + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P332\" style=\"text-indent: 0 !important;\" title=\"4P332\"&gt;\n                        para algum &lt;!-- MATH\n                         $C \\in \\mathbb{R}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.73ex; vertical-align: -0.18ex; \" src=\"img/img2005.svg\" alt=\"$C \\in \\mathbb{R}$\"&gt;&lt;/span&gt;, constante de integração.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P333\" title=\"4P333\"&gt;\n                        A integral do primeiro membro pode ser determinada por\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P334\" title=\"4P334\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.20ex; vertical-align: -2.07ex; \" src=\"img/img2006.svg\" alt=\"$\\displaystyle \\int \\frac{v'}{(1 + \\lambda^{2}v^{2})} dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.60ex; vertical-align: -2.07ex; \" src=\"img/img2007.svg\" alt=\"$\\displaystyle = \\int \\frac{\\frac{dv}{dt}}{(1 + \\lambda^{2}v^{2})} dt$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img2008.svg\" alt=\"$\\displaystyle = \\int \\frac{1}{(1 + \\lambda^{2}v^{2})} dv = \\frac{1}{\\lambda} {\\mathrm {tg}}^{-1}(\\lambda v),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"4P335\" style=\"text-indent: 0 !important;\" title=\"4P335\"&gt;\n                        donde segue que\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        {\\mathrm  {tg}}^{-1}(\\lambda v) = -\\lambda gt + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P336\" title=\"4P336\"&gt;\n                        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2009.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}(\\lambda v) = -\\lambda gt + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P337\" style=\"text-indent: 0 !important;\" title=\"4P337\"&gt;\n                        para alguma constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;. Usando a condição inicial &lt;!-- MATH\n                         $v(t_{0}) = v_{0}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2000.svg\" alt=\"$v(t_{0}) = v_{0}$\"&gt;&lt;/span&gt;, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        C = {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) + \\lambda gt_{0}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P338\" title=\"4P338\"&gt;\n                        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2010.svg\" alt=\"$\\displaystyle C = {\\mathrm {tg}}^{-1}(\\lambda v_{0}) + \\lambda gt_{0}. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P339\" title=\"4P339\"&gt;\n                        Temos portanto que a velocidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\"&gt;&lt;/span&gt; é dada implicitamente por\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P340\" title=\"4P340\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2011.svg\" alt=\"$\\displaystyle {\\mathrm {tg}}^{-1}(\\lambda v)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2012.svg\" alt=\"$\\displaystyle = -\\lambda gt + {\\mathrm {tg}}^{-1}(\\lambda v_{0}) + \\lambda gt_{0},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2013.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}^{-1}(\\lambda v_{0}) - \\lambda g(t-t_{0}).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"4P341\" title=\"4P341\"&gt;\n                        Aplicando tangente em ambos os membros e usando a identidade da soma de arcos para a tangente, temos\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P342\" title=\"4P342\"&gt;&lt;table&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2014.svg\" alt=\"$\\displaystyle \\lambda v = \\lambda v(t)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2015.svg\" alt=\"$\\displaystyle = {\\mathrm {tg}}( {\\mathrm {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0}) )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img2016.svg\" alt=\"$\\displaystyle = \\frac{\\lambda v_{0} - {\\mathrm {tg}}(\\lambda g(t-t_{0}))}{1 + \\lambda v_{0}{\\mathrm {tg}}(\\lambda g(t-t_{0}))},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                        &lt;a id=\"eqvdec\"&gt;&lt;/a&gt;(&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;14&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"4P343\" style=\"text-indent: 0 !important;\"title=\"4P343\"&gt;\n                        para qualquer &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.46ex; \" src=\"img/img2017.svg\" alt=\"$t &gt; t_{0}$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P344\" title=\"4P344\"&gt;\n                        O movimento de ascendência do objeto é então dado em termos da função tangente. Sabemos que este movimento deve\n                        obrigatoriamente cessar na ausência de outras forças. Para ser mais preciso, observe que conforme &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt; cresce, a equação\n                        (&lt;a href=\"#eqvdec\"&gt;4.14&lt;/a&gt;) nos diz que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\"&gt;&lt;/span&gt; decresce. Como a função tangente vai para o infinito continuamente quando o argumento se\n                        aproxima de &lt;!-- MATH\n                         $\\frac{\\pi}{2}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img42.svg\" alt=\"$\\frac{\\pi}{2}$\"&gt;&lt;/span&gt;, então existe um tempo &lt;!-- MATH\n                         $t_{1} &gt; t_{0}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.46ex; \" src=\"img/img2018.svg\" alt=\"$t_{1} &gt; t_{0}$\"&gt;&lt;/span&gt; de forma que o numerador de (&lt;a href=\"#eqvdec\"&gt;4.14&lt;/a&gt;) se anula.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P345\" title=\"4P345\"&gt;\n                        Este tempo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2019.svg\" alt=\"$t_{1}$\"&gt;&lt;/span&gt;, é o tempo em que o objeto atinge a velocidade &lt;!-- MATH\n                         $v(t_{1}) = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2020.svg\" alt=\"$v(t_{1}) = 0$\"&gt;&lt;/span&gt; e começa o movimento de queda livre,\n                        de volta ao solo. Colocando &lt;!-- MATH\n                         $\\lambda v_{0} - {\\mathrm  {tg}}(\\lambda g(t_{1}-t_{0})) = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2021.svg\" alt=\"$\\lambda v_{0} - {\\mathrm {tg}}(\\lambda g(t_{1}-t_{0})) = 0$\"&gt;&lt;/span&gt;, podemos facilmente verificar que\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        t_{1} = \\frac{1}{\\lambda g} {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) + t_{0}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P346\" title=\"4P346\"&gt;\n                        &lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2022.svg\" alt=\"$\\displaystyle t_{1} = \\frac{1}{\\lambda g} {\\mathrm {tg}}^{-1}(\\lambda v_{0}) + t_{0}. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P347\" title=\"4P347\"&gt;\n                        A altura &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2023.svg\" alt=\"$y(t_{1})$\"&gt;&lt;/span&gt; que o objeto atinge, antes de começar a cair, também pode ser determinada. Como sabemos, a\n                        velocidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1989.svg\" alt=\"$v(t)$\"&gt;&lt;/span&gt; do objeto, é dada em termos da sua posição &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1988.svg\" alt=\"$y(t)$\"&gt;&lt;/span&gt; pela igualdade\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        v(t) = \\frac{dy}{dt},\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P348\" title=\"4P348\"&gt;\n                        &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img2024.svg\" alt=\"$\\displaystyle v(t) = \\frac{dy}{dt}, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P349\" style=\"text-indent: 0 !important;\" title=\"4P349\"&gt;\n                        e portanto\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t) = \\int v(t) dt + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P350\" title=\"4P350\"&gt;\n                        &lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img2025.svg\" alt=\"$\\displaystyle y(t) = \\int v(t) dt + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P351\" style=\"text-indent: 0 !important;\" title=\"4P351\"&gt;\n                        para alguma constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;. Substituindo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1989.svg\" alt=\"$v(t)$\"&gt;&lt;/span&gt; temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t) = \\int \\frac{1}{\\lambda} {\\mathrm  {tg}}( {\\mathrm  {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0}) ) dt + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P352\" title=\"4P352\"&gt;\n                        &lt;img style=\"height: 4.89ex; vertical-align: -1.92ex; \" src=\"img/img2026.svg\" alt=\"$\\displaystyle y(t) = \\int \\frac{1}{\\lambda} {\\mathrm {tg}}( {\\mathrm {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0}) ) dt + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P353\" style=\"text-indent: 0 !important;\" title=\"4P353\"&gt;\n                        e fazendo a mudança de variáveis &lt;!-- MATH\n                         $s = {\\mathrm  {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0})$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2027.svg\" alt=\"$s = {\\mathrm {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0})$\"&gt;&lt;/span&gt;, temos &lt;!-- MATH\n                         $\\frac{ds}{dt} = -\\lambda g$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2028.svg\" alt=\"$\\frac{ds}{dt} = -\\lambda g$\"&gt;&lt;/span&gt;,\n                        e assim\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P354\" title=\"4P354\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2029.svg\" alt=\"$\\displaystyle y(t)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.89ex; vertical-align: -1.92ex; \" src=\"img/img2030.svg\" alt=\"$\\displaystyle = \\frac{1}{\\lambda} \\int {\\mathrm {tg}}( {\\mathrm {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0}) ) dt + C$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2031.svg\" alt=\"$\\displaystyle = \\frac{-1}{\\lambda^{2}g} \\int {\\mathrm {tg}}s ds + C$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2032.svg\" alt=\"$\\displaystyle = \\frac{-1}{\\lambda^{2}g} \\int \\frac{{\\mathrm {sen}}s}{\\cos s} ds + C = \\frac{-1}{\\lambda^{2}g} \\ln\\vert\\cos s\\vert + C.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"4P355\" title=\"4P355\"&gt;\n                        Observe que nesta etapa, estamos considerando que &lt;!-- MATH\n                         $t_{0} \\leq t \\leq t_{1}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.89ex; vertical-align: -0.46ex; \" src=\"img/img2033.svg\" alt=\"$t_{0} \\leq t \\leq t_{1}$\"&gt;&lt;/span&gt;. Nestes termos,\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        0 &lt; \\lambda g(t-t_{0}) &lt; \\lambda g(t_{1}-t_{0}),\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P356\" title=\"4P356\"&gt;\n                        &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2034.svg\" alt=\"$\\displaystyle 0 &lt; \\lambda g(t-t_{0}) &lt; \\lambda g(t_{1}-t_{0}), $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P357\" style=\"text-indent: 0 !important;\" title=\"4P357\"&gt;\n                        e lembrando que &lt;!-- MATH\n                         $\\lambda g(t_{1}-t_{0}) = {\\mathrm  {tg}}^{-1}(\\lambda v_{0})$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2035.svg\" alt=\"$\\lambda g(t_{1}-t_{0}) = {\\mathrm {tg}}^{-1}(\\lambda v_{0})$\"&gt;&lt;/span&gt;, temos que\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        0 &lt; {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) - \\lambda g(t-t_{0}) &lt; {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) &lt; \\frac{\\pi}{2}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P358\" title=\"4P358\"&gt;\n                        &lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img2036.svg\" alt=\"$\\displaystyle 0 &lt; {\\mathrm {tg}}^{-1}(\\lambda v_{0}) - \\lambda g(t-t_{0}) &lt; {\\mathrm {tg}}^{-1}(\\lambda v_{0}) &lt; \\frac{\\pi}{2}. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P359\" title=\"4P359\"&gt;\n                        Isto significa que &lt;!-- MATH\n                         $0 &lt; s &lt; \\frac{\\pi}{2}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.83ex; \" src=\"img/img2037.svg\" alt=\"$0 &lt; s &lt; \\frac{\\pi}{2}$\"&gt;&lt;/span&gt; e neste intervalo temos que &lt;!-- MATH\n                         $\\cos s &gt; 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img2038.svg\" alt=\"$\\cos s &gt; 0$\"&gt;&lt;/span&gt;. Podemos portanto descartar o\n                        módulo no logaritmo. Segue que\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P360\" title=\"4P360\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img2039.svg\" alt=\"$\\displaystyle y$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2040.svg\" alt=\"$\\displaystyle = \\frac{-1}{\\lambda^{2}g} \\ln(\\cos s) + C$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2041.svg\" alt=\"$\\displaystyle = \\frac{-1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0}) ) \\right)+ C,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"4P361\" style=\"text-indent: 0 !important;\" title=\"4P361\"&gt;\n                        para alguma constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;. Para determinar a constante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;, usamos a condição inicial &lt;!-- MATH\n                         $y(t_{0}) = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2042.svg\" alt=\"$y(t_{0}) = 0$\"&gt;&lt;/span&gt;, que\n                        nos fornece &lt;!-- MATH\n                         $C = \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos\\left( {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) \\right) \\right)$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.21ex; vertical-align: -1.23ex; \" src=\"img/img2043.svg\" alt=\"$C = \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos\\left( {\\mathrm {tg}}^{-1}(\\lambda v_{0}) \\right) \\right)$\"&gt;&lt;/span&gt;. Temos assim, que\n                        a altura do objeto em um determinado tempo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt; com &lt;!-- MATH\n                         $t_{0} \\leq t \\leq t_{1}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.89ex; vertical-align: -0.46ex; \" src=\"img/img2033.svg\" alt=\"$t_{0} \\leq t \\leq t_{1}$\"&gt;&lt;/span&gt;, é dada por\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t) = \\frac{-1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm  {tg}}^{-1}(\\lambda v_{0})-\\lambda g(t-t_{0}) ) \\right) + \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) ) \\right),\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P362\" title=\"4P362\"&gt;\n                        &lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2044.svg\" alt=\"$\\displaystyle y(t) = \\frac{-1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm {tg}}^{-1...\n                        ...1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm {tg}}^{-1}(\\lambda v_{0}) ) \\right), $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P363\" style=\"text-indent: 0 !important;\" title=\"4P363\"&gt;\n                        sendo portanto a altura máxima que o objeto atinge\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t_{1}) = \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) ) \\right).\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P364\" title=\"4P364\"&gt;\n                        &lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2045.svg\" alt=\"$\\displaystyle y(t_{1}) = \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm {tg}}^{-1}(\\lambda v_{0}) ) \\right). $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P365\" title=\"4P365\"&gt;\n                        Agora, vamos seguir o estudo quando &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2046.svg\" alt=\"$t &gt; t_{1}$\"&gt;&lt;/span&gt;. No instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2019.svg\" alt=\"$t_{1}$\"&gt;&lt;/span&gt;, o objeto atinge a altura máxima e começa então o\n                        movimento de descida do objeto conhecido como movimento de queda livre. O problema da queda livre pode ser totalmente\n                        desvinculado do que esquematizamos até agora. Um objeto pode cair em queda livre sem ter sido necessariamente\n                        arremessado para cima. Um exemplo disto é um paraquedista que salta de um avião.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P366\" title=\"4P366\"&gt;\n                        A equação de descida do objeto é um pouco diferente pois agora a força de atrito age no mesmo sentido do referencial. A\n                        equação das forças é agora dada por\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        ma = F = p + r = -mg + kv^{2},\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P367\" title=\"4P367\"&gt;\n                        &lt;img style=\"height: 2.53ex; vertical-align: -0.50ex; \" src=\"img/img2047.svg\" alt=\"$\\displaystyle ma = F = p + r = -mg + kv^{2}, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P368\" style=\"text-indent: 0 !important;\" title=\"4P368\"&gt;\n                        e, portanto, a equação diferencial é dada por\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P369\" title=\"4P369\"&gt;&lt;a id=\"eqpquedas\"&gt;&lt;/a&gt;&lt;!-- MATH\n                         \\begin{equation}\n                        v' = a = g\\left( \\tfrac{k}{gm}v^{2} - 1 \\right) = g(\\lambda^{2}v^{2}-1),\n                        \\end{equation}\n                         --&gt;\n                        &lt;table &gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img2048.svg\" alt=\"$\\displaystyle v' = a = g\\left( \\tfrac{k}{gm}v^{2} - 1 \\right) = g(\\lambda^{2}v^{2}-1),$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                            (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;15&lt;/span&gt;)  &lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt; &lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"4P370\" style=\"text-indent: 0 !important;\" title=\"4P370\"&gt;\n                        sujeita à condição inicial &lt;!-- MATH\n                         $v(t_{1}) = v_{1}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2049.svg\" alt=\"$v(t_{1}) = v_{1}$\"&gt;&lt;/span&gt;. No caso do corpo ser arremessado verticalmente para cima, temos que\n                        &lt;!-- MATH\n                         $v(t_{1}) = v_{1} = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2050.svg\" alt=\"$v(t_{1}) = v_{1} = 0$\"&gt;&lt;/span&gt;. Por uma questão de simplicidade substituímos o termo &lt;!-- MATH\n                         $\\frac{k}{gm}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.10ex; vertical-align: -1.12ex; \" src=\"img/img2051.svg\" alt=\"$\\frac{k}{gm}$\"&gt;&lt;/span&gt; por &lt;!-- MATH\n                         $\\lambda^{2}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.02ex; vertical-align: -0.10ex; \" src=\"img/img2052.svg\" alt=\"$\\lambda^{2}$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P371\" title=\"4P371\"&gt;\n                        Temos portanto um problema de valor inicial\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\left\\{ \\begin{array}{l} v' = g(\\lambda^{2}v^{2}-1) \\\\v(t_{1}) = v_{1}. \\end{array} \\right.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P372\" title=\"4P372\"&gt;\n                        &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img2053.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l} v' = g(\\lambda^{2}v^{2}-1) \\\\ v(t_{1}) = v_{1}. \\end{array} \\right. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P373\" title=\"4P373\"&gt;\n                        Reescrevendo a equação diferencial obtemos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\frac{1}{(\\lambda^{2}v^{2}-1)} v' = g,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P374\" title=\"4P374\"&gt;\n                        &lt;img style=\"height: 5.04ex; vertical-align: -2.07ex; \" src=\"img/img2054.svg\" alt=\"$\\displaystyle \\frac{1}{(\\lambda^{2}v^{2}-1)} v' = g, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P375\" style=\"text-indent: 0 !important;\" title=\"4P375\"&gt;\n                        e integrando em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, com a mudança de variáveis &lt;!-- MATH\n                         $u =\\lambda v$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2055.svg\" alt=\"$u =\\lambda v$\"&gt;&lt;/span&gt;, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        gt = \\int \\frac{1}{\\lambda^{2}v^{2}-1} v' dt = \\int \\frac{1}{\\lambda^{2}v^{2}-1} dv = -\\frac{1}{\\lambda} \\int \\frac{1}{1-u^{2}} du.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P376\" title=\"4P376\"&gt;\n                        &lt;img style=\"height: 4.89ex; vertical-align: -1.92ex; \" src=\"img/img2056.svg\" alt=\"$\\displaystyle gt = \\int \\frac{1}{\\lambda^{2}v^{2}-1} v' dt = \\int \\frac{1}{\\lambda^{2}v^{2}-1} dv = -\\frac{1}{\\lambda} \\int \\frac{1}{1-u^{2}} du. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P377\" title=\"4P377\"&gt;\n                        De acordo com as fórmulas de derivação para as funções trigonométricas hiperbólicas inversas (ver tabela &lt;a href=\"/trigonometria-hiperbolica/igualdades-exponenciais-e-logaritmicas#tabdfhi\"&gt;3.1&lt;/a&gt;)\n                        temos que a integral do último membro é igual a &lt;!-- MATH\n                         ${\\mathrm {tgh}}^{-1} u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1055.svg\" alt=\"${\\mathrm {tgh}}^{-1} u$\"&gt;&lt;/span&gt; para &lt;!-- MATH\n                         $u \\in (-1,1)$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img552.svg\" alt=\"$u \\in (-1,1)$\"&gt;&lt;/span&gt;, ou igual a &lt;!-- MATH\n                         ${\\mathrm{ctgh}}^{-1} u$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.59ex; vertical-align: -0.58ex; \" src=\"img/img1056.svg\" alt=\"${\\mathrm{ctgh}}^{-1} u$\"&gt;&lt;/span&gt; se &lt;!-- MATH\n                         $u \\in\n                        (-\\infty,-1) \\cup (1,\\infty)$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img583.svg\" alt=\"$u \\in\n                        (-\\infty,-1) \\cup (1,\\infty)$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P378\" title=\"4P378\"&gt;\n                        Observemos que não há a possibilidade de que &lt;!-- MATH\n                         $u = \\lambda v = \\pm 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.83ex; vertical-align: -0.27ex; \" src=\"img/img2057.svg\" alt=\"$u = \\lambda v = \\pm 1$\"&gt;&lt;/span&gt;, em virtude de que a força de aceleração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.81ex; vertical-align: -0.10ex; \" src=\"img/img2058.svg\" alt=\"$v'$\"&gt;&lt;/span&gt;\n                        nunca se anula (a menos que não haja gravidade) e, portanto, o lado direito da igualdade (&lt;a href=\"#eqpquedas\"&gt;4.15&lt;/a&gt;) também nunca\n                        se anula.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P379\" title=\"4P379\"&gt;\n                        Resta que ou &lt;!-- MATH\n                         $\\lambda^{2} v^{2} &gt; 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.14ex; \" src=\"img/img2059.svg\" alt=\"$\\lambda^{2} v^{2} &gt; 1$\"&gt;&lt;/span&gt; ou &lt;!-- MATH\n                         $\\lambda^{2} v^{2} &lt; 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.14ex; \" src=\"img/img2060.svg\" alt=\"$\\lambda^{2} v^{2} &lt; 1$\"&gt;&lt;/span&gt;. Isto será uma decorrência da velocidade inicial de\n                        queda &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.51ex; vertical-align: -0.45ex; \" src=\"img/img2061.svg\" alt=\"$v_{1}$\"&gt;&lt;/span&gt;. Se &lt;!-- MATH\n                         $v_{1}^{2} = ( v(t_{1}) )^{2} &gt; \\frac{1}{\\lambda^{2}}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.92ex; vertical-align: -0.94ex; \" src=\"img/img2062.svg\" alt=\"$v_{1}^{2} = ( v(t_{1}) )^{2} &gt; \\frac{1}{\\lambda^{2}}$\"&gt;&lt;/span&gt; então esta desigualdade se mantem para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2046.svg\" alt=\"$t &gt; t_{1}$\"&gt;&lt;/span&gt; e se &lt;!-- MATH\n                         $v_{1}^{2} &lt; \\frac{1}{\\lambda^{2}}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.92ex; vertical-align: -0.94ex; \" src=\"img/img2063.svg\" alt=\"$v_{1}^{2} &lt; \\frac{1}{\\lambda^{2}}$\"&gt;&lt;/span&gt; então isto se mantem para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2046.svg\" alt=\"$t &gt; t_{1}$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P380\" title=\"4P380\"&gt;\n                        Para não sobrecarregar (ainda mais) o texto, vamos escolher uma das duas situações observadas acima. Para ficar\n                        consistente com início da seção, isto é, o arremesso vertical e o instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2019.svg\" alt=\"$t_{1}$\"&gt;&lt;/span&gt; no qual &lt;!-- MATH\n                         $v_{1} = v(t_{1}) = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2064.svg\" alt=\"$v_{1} = v(t_{1}) = 0$\"&gt;&lt;/span&gt;,\n                        escolhemos o caso em que &lt;!-- MATH\n                         $\\lambda^{2} v^{2} &lt; 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.06ex; vertical-align: -0.14ex; \" src=\"img/img2060.svg\" alt=\"$\\lambda^{2} v^{2} &lt; 1$\"&gt;&lt;/span&gt;. Assumindo esta condição, temos que &lt;!-- MATH\n                         $u = \\lambda v \\in (-1,1)$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2065.svg\" alt=\"$u = \\lambda v \\in (-1,1)$\"&gt;&lt;/span&gt; e a\n                        solução da equação diferencial será dada implicitamente por\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        gt = -\\frac{1}{\\lambda} \\int \\frac{1}{1-u^{2}} du = -\\frac{1}{\\lambda} {\\mathrm {tgh}}^{-1} u + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P381\" title=\"4P381\"&gt;\n                        &lt;img style=\"height: 4.89ex; vertical-align: -1.92ex; \" src=\"img/img2066.svg\" alt=\"$\\displaystyle gt = -\\frac{1}{\\lambda} \\int \\frac{1}{1-u^{2}} du = -\\frac{1}{\\lambda} {\\mathrm {tgh}}^{-1} u + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P382\" style=\"text-indent: 0 !important;\" title=\"4P382\"&gt;\n                        e voltando à variável &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\"&gt;&lt;/span&gt;,\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\lambda gt = -{\\mathrm {tgh}}^{-1}(\\lambda v) + C.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P383\" title=\"4P383\"&gt;\n                        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2067.svg\" alt=\"$\\displaystyle \\lambda gt = -{\\mathrm {tgh}}^{-1}(\\lambda v) + C. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P384\" title=\"4P384\"&gt;\n                        A constante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;, calculada pela condição inicial &lt;!-- MATH\n                         $v(t_{1}) = v_{1}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2049.svg\" alt=\"$v(t_{1}) = v_{1}$\"&gt;&lt;/span&gt;, é\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        C = \\lambda g t_{1} + {\\mathrm {tgh}}^{-1}(\\lambda v_{1}),\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P385\" title=\"4P385\"&gt;\n                        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2068.svg\" alt=\"$\\displaystyle C = \\lambda g t_{1} + {\\mathrm {tgh}}^{-1}(\\lambda v_{1}), $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P386\" style=\"text-indent: 0 !important;\" title=\"4P386\"&gt;\n                        o que nos leva a\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\lambda gt = -{\\mathrm {tgh}}^{-1}(\\lambda v) + \\lambda g t_{1} + {\\mathrm {tgh}}^{-1}(\\lambda v_{1}).\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P387\" title=\"4P387\"&gt;\n                        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2069.svg\" alt=\"$\\displaystyle \\lambda gt = -{\\mathrm {tgh}}^{-1}(\\lambda v) + \\lambda g t_{1} + {\\mathrm {tgh}}^{-1}(\\lambda v_{1}). $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P388\" title=\"4P388\"&gt;\n                        Com o intuito de isolar a velocidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\"&gt;&lt;/span&gt;, reescrevemos a igualdade na forma\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        {\\mathrm {tgh}}^{-1}(\\lambda v) = {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1}),\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P389\" title=\"4P389\"&gt;\n                        &lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2070.svg\" alt=\"$\\displaystyle {\\mathrm {tgh}}^{-1}(\\lambda v) = {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1}), $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P390\" style=\"text-indent: 0 !important;\" title=\"4P390\"&gt;\n                        e aplicamos tangente hiperbólica em ambos os membros, obtendo\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\lambda v = {\\mathrm {tgh}}({\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1})) = \\frac{\\lambda v_{1} - {\\mathrm {tgh}}(\\lambda g(t-t_{1}))}{1-\\lambda v_{1}{\\mathrm {tgh}}(\\lambda g(t-t_{1}))}.\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P391\" title=\"4P391\"&gt;\n                        &lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img2071.svg\" alt=\"$\\displaystyle \\lambda v = {\\mathrm {tgh}}({\\mathrm {tgh}}^{-1}(\\lambda v_{1}) -...\n                        ...gh}}(\\lambda g(t-t_{1}))}{1-\\lambda v_{1}{\\mathrm {tgh}}(\\lambda g(t-t_{1}))}. $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P392\" title=\"4P392\"&gt;\n                        A velocidade de queda do objeto é dada portanto em termos da função tangente hiperbólica. Vamos analisar o\n                        comportamento da velocidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\"&gt;&lt;/span&gt; quando &lt;!-- MATH\n                         $t \\to \\infty$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;. Observe que na prática não podemos considerar &lt;!-- MATH\n                         $t \\to \\infty$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;,\n                        porque certamente o objeto atinge o solo em um tempo finito. Mas esta análise nos trará boas ideias sobre a velocidade\n                        terminal, isto é, a velocidade aproximada com que o objeto atinge o solo.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P393\" title=\"4P393\"&gt;\n                        Lembremos que &lt;!-- MATH\n                         ${\\mathrm {tgh}}u \\to 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img2073.svg\" alt=\"${\\mathrm {tgh}}u \\to 1$\"&gt;&lt;/span&gt;, quando &lt;!-- MATH\n                         $u \\to \\infty$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img863.svg\" alt=\"$u \\to \\infty$\"&gt;&lt;/span&gt; (Ver seção &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-hiperbolicas#secfunchip\"&gt;2.3&lt;/a&gt;). Assim, quando &lt;!-- MATH\n                         $t \\to \\infty$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        \\lambda v = \\frac{\\lambda v_{1} - {\\mathrm {tgh}}(2\\lambda g(t-t_{1}))}{1-\\lambda v_{1} {\\mathrm {tgh}}(2\\lambda g(t-t_{1}))} \\to \\frac{\\lambda v_{1} - 1}{1-\\lambda v_{1}} = -1,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P394\" title=\"4P394\"&gt;\n                        &lt;img style=\"height: 5.16ex; vertical-align: -2.07ex; \" src=\"img/img2074.svg\" alt=\"$\\displaystyle \\lambda v = \\frac{\\lambda v_{1} - {\\mathrm {tgh}}(2\\lambda g(t-t_...\n                        ...gh}}(2\\lambda g(t-t_{1}))} \\to \\frac{\\lambda v_{1} - 1}{1-\\lambda v_{1}} = -1, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P395\" style=\"text-indent: 0 !important;\" title=\"4P395\"&gt;\n                        donde &lt;!-- MATH\n                         $v \\to -\\frac{1}{\\lambda} = -\\sqrt{\\frac{mg}{k}}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.07ex; vertical-align: -1.44ex; \" src=\"img/img2075.svg\" alt=\"$v \\to -\\frac{1}{\\lambda} = -\\sqrt{\\frac{mg}{k}}$\"&gt;&lt;/span&gt;. O sinal negativo decorre do fato de que a velocidade do objeto\n                        é um vetor que aponta em sentido contrário ao referencial escolhido.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P396\" title=\"4P396\"&gt;\n                        Supondo que o objeto em questão fosse um paraquedista com paraquedas aberto, temos que a velocidade terminal do\n                        paraquedista (em módulo) é &lt;!-- MATH\n                         $v_{t} = \\frac{1}{\\lambda} = \\sqrt{\\frac{mg}{k}} = \\sqrt{\\frac{2p}{\\rho A\\delta}}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.10ex; vertical-align: -1.44ex; \" src=\"img/img2076.svg\" alt=\"$v_{t} = \\frac{1}{\\lambda} = \\sqrt{\\frac{mg}{k}} = \\sqrt{\\frac{2p}{\\rho A\\delta}}$\"&gt;&lt;/span&gt;, sendo\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img2077.svg\" alt=\"$p$\"&gt;&lt;/span&gt; o peso do paraquedista, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img1993.svg\" alt=\"$\\rho$\"&gt;&lt;/span&gt; a densidade do ar e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\"&gt;&lt;/span&gt; a área do paraquedas.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P397\" title=\"4P397\"&gt;\n                        Vemos então que quanto maior for o peso do paraquedista, maior a velocidade terminal. Também, quanto maiores forem a\n                        área do paraquedas ou a densidade do ar, menor a velocidade terminal.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P398\" title=\"4P398\"&gt;\n                        Podemos também determinar o tempo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.86ex; vertical-align: -0.44ex; \" src=\"img/img2078.svg\" alt=\"$t_{2}$\"&gt;&lt;/span&gt; que o objeto leva para atingir o solo novamente. Este tempo é exatamente o\n                        tempo em que &lt;!-- MATH\n                         $y(t_{2}) = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2079.svg\" alt=\"$y(t_{2}) = 0$\"&gt;&lt;/span&gt;. O problema momentâneo é que não temos ainda uma identidade para a posição &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img71.svg\" alt=\"$y$\"&gt;&lt;/span&gt; do objeto,\n                        no momento de queda. Vamos determinar esta igualdade. Como feito anteriormente\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t) = \\int v(t) dt + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P399\" title=\"4P399\"&gt;\n                        &lt;img style=\"height: 4.87ex; vertical-align: -1.92ex; \" src=\"img/img2025.svg\" alt=\"$\\displaystyle y(t) = \\int v(t) dt + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P400\" style=\"text-indent: 0 !important;\" title=\"4P400\"&gt;\n                        para alguma constante de integração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img133.svg\" alt=\"$C$\"&gt;&lt;/span&gt;, que será determinada pela condição inicial &lt;!-- MATH\n                         $y(t_{1}) = h$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2080.svg\" alt=\"$y(t_{1}) = h$\"&gt;&lt;/span&gt; a altura em que o\n                        objeto foi solto em queda livre no instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.87ex; vertical-align: -0.45ex; \" src=\"img/img2019.svg\" alt=\"$t_{1}$\"&gt;&lt;/span&gt;. No caso de o objeto ter sido arremessado verticalmente, então\n                        lembremos que esta condição será &lt;!-- MATH\n                         $y(t_{t}) = h = \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm  {tg}}^{-1}(\\lambda v_{0}) )\n                        \\right)$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.21ex; vertical-align: -1.23ex; \" src=\"img/img2081.svg\" alt=\"$y(t_{t}) = h = \\frac{1}{\\lambda^{2}g} \\ln\\left( \\cos( {\\mathrm {tg}}^{-1}(\\lambda v_{0}) )\n                        \\right)$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;\n                        &lt;p class=\" unidade\" id=\"4P401\" title=\"4P401\"&gt;\n                        Substituindo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1989.svg\" alt=\"$v(t)$\"&gt;&lt;/span&gt; na equação integral, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t) = \\frac{1}{\\lambda} \\int {\\mathrm {tgh}}({\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1})) dt + C,\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P402\" title=\"4P402\"&gt;\n                        &lt;img style=\"height: 4.89ex; vertical-align: -1.92ex; \" src=\"img/img2082.svg\" alt=\"$\\displaystyle y(t) = \\frac{1}{\\lambda} \\int {\\mathrm {tgh}}({\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1})) dt + C, $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P403\" style=\"text-indent: 0 !important;\" title=\"4P403\"&gt;\n                        e fazendo a mudança de variáveis &lt;!-- MATH\n                         $s = {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1})$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.63ex; vertical-align: -0.62ex; \" src=\"img/img2083.svg\" alt=\"$s = {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1})$\"&gt;&lt;/span&gt;, temos &lt;!-- MATH\n                         $\\frac{ds}{dt} = -\\lambda\n                        g$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2028.svg\" alt=\"$\\frac{ds}{dt} = -\\lambda g$\"&gt;&lt;/span&gt; e, assim,\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P404\" title=\"4P404\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2029.svg\" alt=\"$\\displaystyle y(t)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2084.svg\" alt=\"$\\displaystyle = -\\frac{1}{\\lambda^{2}g} \\int {\\mathrm {tgh}}s ds + C$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2085.svg\" alt=\"$\\displaystyle = -\\frac{1}{\\lambda^{2}g} \\int \\frac{{\\mathrm{senh}}s}{\\cosh s} ds + C$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2086.svg\" alt=\"$\\displaystyle = -\\frac{1}{\\lambda^{2}g} \\ln( \\cosh s )+ C$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2087.svg\" alt=\"$\\displaystyle = -\\frac{1}{\\lambda^{2}g} \\ln\\left(\\cosh( {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1}) ) \\right) + C.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        \n                        \n                        &lt;p class=\" unidade\" id=\"4P405\" title=\"4P405\"&gt;\n                        Substituindo a condição &lt;!-- MATH\n                         $y(t_{1}) = h$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2080.svg\" alt=\"$y(t_{1}) = h$\"&gt;&lt;/span&gt;, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        C = h + \\frac{1}{\\lambda^{2}g} \\ln \\cosh( {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) ),\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P406\" title=\"4P406\"&gt;\n                        &lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2088.svg\" alt=\"$\\displaystyle C = h + \\frac{1}{\\lambda^{2}g} \\ln \\cosh( {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) ), $\"&gt;\n                        &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P407\" style=\"text-indent: 0 !important;\" title=\"4P407\"&gt;\n                        e, assim,\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        y(t) = h + \\frac{1}{\\lambda^{2}g} \\ln\\left(\\cosh( {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) ) \\right)\n                                  - \\frac{1}{\\lambda^{2}g} \\ln\\left(\\cosh( {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1}) ) \\right). \n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P408\" title=\"4P408\"&gt;\n                        &lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2089.svg\" alt=\"$\\displaystyle y(t) = h + \\frac{1}{\\lambda^{2}g} \\ln\\left(\\cosh( {\\mathrm {tgh}}...\n                        ...eft(\\cosh( {\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\lambda g(t-t_{1}) ) \\right). $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P409\" title=\"4P409\"&gt;\n                        Reorganizando temos que\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P410\" title=\"4P410\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.53ex; vertical-align: -0.50ex; \" src=\"img/img2090.svg\" alt=\"$\\displaystyle \\lambda^{2} gy$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.74ex; vertical-align: -2.31ex; \" src=\"img/img2091.svg\" alt=\"$\\displaystyle = \\lambda^{2}gh + \\ln\\frac{\\cosh({\\mathrm {tgh}}^{-1}(\\lambda v_{1}))}{\\cosh({\\mathrm {tgh}}^{-1}(\\lambda v_{1})-\\lambda g(t-t_{1}))}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.74ex; vertical-align: -2.31ex; \" src=\"img/img2092.svg\" alt=\"$\\displaystyle = \\lambda^{2}gh - \\ln\\frac{\\cosh({\\mathrm {tgh}}^{-1}(\\lambda v_{1})-\\lambda g(t-t_{1}))}{\\cosh({\\mathrm {tgh}}^{-1}(\\lambda v_{1}))},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"4P411\" style=\"text-indent: 0 !important;\" title=\"4P411\"&gt;\n                        e calculando o tempo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.86ex; vertical-align: -0.44ex; \" src=\"img/img2078.svg\" alt=\"$t_{2}$\"&gt;&lt;/span&gt;, para que &lt;!-- MATH\n                         $y(t_{2}) = 0$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2079.svg\" alt=\"$y(t_{2}) = 0$\"&gt;&lt;/span&gt;, temos\n                        &lt;!-- MATH\n                         \\begin{displaymath}\n                        t_{2} = t_{1} + \\frac{1}{\\lambda g}\\left({\\mathrm {tgh}}^{-1}(\\lambda v_{1}) - \\cosh^{-1} \\left(e^{\\lambda^{2}gh}\n                        \\cosh({\\mathrm {tgh}}^{-1}(\\lambda v_{1})) \\right) \\right).\n                        \\end{displaymath}\n                         --&gt;\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P412\" title=\"4P412\"&gt;\n                        &lt;img style=\"height: 4.92ex; vertical-align: -1.95ex; \" src=\"img/img2093.svg\" alt=\"$\\displaystyle t_{2} = t_{1} + \\frac{1}{\\lambda g}\\left({\\mathrm {tgh}}^{-1}(\\la...\n                        ...^{\\lambda^{2}gh}\n                        \\cosh({\\mathrm {tgh}}^{-1}(\\lambda v_{1})) \\right) \\right).\n                        $\"&gt;\n                        &lt;/div&gt;\n                        \n                        &lt;p class=\" unidade\" id=\"4P413\" title=\"4P413\"&gt;\n                        Para finalizar, observamos que se tivéssemos escolhido &lt;!-- MATH\n                         $\\lambda^{2} v_{1}^{2} &gt; 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.67ex; vertical-align: -0.74ex; \" src=\"img/img2094.svg\" alt=\"$\\lambda^{2} v_{1}^{2} &gt; 1$\"&gt;&lt;/span&gt;, então a velocidade seria dada por\n                        &lt;/p&gt;\n                        &lt;div class=\"mathdisplay unidade\" id=\"4P414\" title=\"4P414\"&gt;&lt;table class=\"equation\"&gt;\n                        &lt;tbody&gt;&lt;tr&gt;\n                        &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2095.svg\" alt=\"$\\displaystyle \\lambda v$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.62ex; \" src=\"img/img2096.svg\" alt=\"$\\displaystyle = {\\mathrm{ctgh}}( {\\mathrm{ctgh}}^{-1}(\\lambda v_{1}) -\\lambda g(t-t_{1}) )$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.74ex; vertical-align: -2.31ex; \" src=\"img/img2097.svg\" alt=\"$\\displaystyle = \\frac{1-{\\mathrm {tgh}}({\\mathrm{ctgh}}^{-1}(\\lambda v_{1})) {\\...\n                        ...h}}({\\mathrm{ctgh}}^{-1}(\\lambda v_{1})) - {\\mathrm {tgh}}(\\lambda g(t-t_{1}))}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;tr&gt;\n                        &lt;td&gt;&nbsp;&lt;/td&gt;\n                        &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.57ex; vertical-align: -2.77ex; \" src=\"img/img2098.svg\" alt=\"$\\displaystyle = \\frac{1- \\frac{1}{\\lambda v_{1}} {\\mathrm {tgh}}(\\lambda g(t-t_...\n                        ...(\\lambda g(t-t_{1})) }{ 1 - \\lambda v_{1} {\\mathrm {tgh}}(\\lambda g(t-t_{1}))},$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                        &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                        &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                        &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                        &lt;p class=\" unidade\" id=\"4P415\" style=\"text-indent: 0 !important;\" title=\"4P415\"&gt;\n                        e sabendo que &lt;!-- MATH\n                         ${\\mathrm {tgh}}u \\to 1$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.13ex; vertical-align: -0.58ex; \" src=\"img/img2073.svg\" alt=\"${\\mathrm {tgh}}u \\to 1$\"&gt;&lt;/span&gt; quando &lt;!-- MATH\n                         $u \\to \\infty$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img863.svg\" alt=\"$u \\to \\infty$\"&gt;&lt;/span&gt;, ainda teríamos &lt;!-- MATH\n                         $v \\to -\\frac{1}{\\lambda} = -\\sqrt{\\frac{mg}{k}}$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.07ex; vertical-align: -1.44ex; \" src=\"img/img2075.svg\" alt=\"$v \\to -\\frac{1}{\\lambda} = -\\sqrt{\\frac{mg}{k}}$\"&gt;&lt;/span&gt;\n                        quando &lt;!-- MATH\n                         $t \\to \\infty$\n                         --&gt;\n                        &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;.\n                        &lt;/p&gt;",
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      "Capítulo 4: Aplicações"
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    "title": "Capítulo 4: Aplicações",
    "section": "4.6 O pêndulo simples",
    "text": "4.6 O pêndulo simples\n\n                            &lt;p class=\" unidade\" id=\"4P416\" title=\"4P416\"&gt;\n                            Um pêndulo consiste de um objeto de massa &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; preso a um suporte horizontal rígido por um fio de comprimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1841.svg\" alt=\"$L$\"&gt;&lt;/span&gt;. O fio\n                            é considerado rígido, inextensível e com massa desprezível. Este objeto é solto de uma posição inicial, onde o fio faz\n                            um ângulo &lt;!-- MATH\n                             $\\theta_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.01ex; vertical-align: -0.46ex; \" src=\"img/img2099.svg\" alt=\"$\\theta_{0}$\"&gt;&lt;/span&gt; com a perpendicular e começa a oscilar em movimento de vai-e-vem. Vamos considerar que\n                            &lt;!-- MATH\n                             $\\theta_{0} &gt; 0$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.01ex; vertical-align: -0.46ex; \" src=\"img/img2100.svg\" alt=\"$\\theta_{0} &gt; 0$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P417\" title=\"4P417\"&gt;\n                            Uma vez solto o pêndulo, o ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt; que o fio faz com a perpendicular, varia com o tempo. Nestes termos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt;\n                            é uma função da variável temporal &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, isto é, &lt;!-- MATH\n                             $\\theta = \\theta(t)$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2101.svg\" alt=\"$\\theta = \\theta(t)$\"&gt;&lt;/span&gt; e, além disso, &lt;!-- MATH\n                             $-\\theta_{0} &lt; \\theta &lt; \\theta_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.01ex; vertical-align: -0.46ex; \" src=\"img/img2102.svg\" alt=\"$-\\theta_{0} &lt; \\theta &lt; \\theta_{0}$\"&gt;&lt;/span&gt;.\n                            A situação pode ser visualizada na figura &lt;a href=\"#figpen\"&gt;4.4&lt;/a&gt;.\n                            &lt;/p&gt;\n                            &lt;div class=\"CENTER\"&gt;&lt;a id=\"figpen\"&gt;&lt;/a&gt;&lt;a id=\"7400\"&gt;&lt;/a&gt;\n                            &lt;table&gt;\n                            &lt;caption class=\"BOTTOM\"&gt;&lt;strong&gt;Figura 4.4:&lt;/strong&gt;\n                            Pêndulo simples.&lt;/caption&gt;\n                            &lt;tbody&gt;&lt;tr&gt;&lt;td&gt;\n                            &lt;div class=\"CENTER\"&gt;\n                            &lt;img src=\"img/pendulo.png\" alt=\"Image pendulo\"&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;\n                            &lt;/tbody&gt;&lt;/table&gt;\n                            &lt;/div&gt;\n                            \n                            &lt;p class=\" unidade\" id=\"4P418\" title=\"4P418\"&gt;\n                            O movimento do objeto se dá em um plano bidimensional e descreve neste plano uma trajetória circular. Fixemos um\n                            sistema coordenado bidimensional nas coordenadas tangencial e radial ao movimento circular. Isto é, um dos eixos é\n                            tangente à trajetória circular enquanto o outro eixo é normal (perpendicular) à trajetória circular.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P419\" title=\"4P419\"&gt;\n                            Sobre este objeto agem a força peso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img1837.svg\" alt=\"$\\vec{p}$\"&gt;&lt;/span&gt;, a força &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt; de tensão com a haste e uma força &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.76ex; vertical-align: -0.10ex; \" src=\"img/img2103.svg\" alt=\"$\\vec{r}$\"&gt;&lt;/span&gt; de atrito\n                            (ou resistência do ar).\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P420\" title=\"4P420\"&gt;\n                            A força peso é decomposta, em termos do ângulo que a haste faz com a vertical, nas componentes tangencial e radial como\n                            &lt;!-- MATH\n                             $\\vec{p} = (-mg {\\mathrm {sen}}\\theta, -mg \\cos \\theta)$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2104.svg\" alt=\"$\\vec{p} = (-mg {\\mathrm {sen}}\\theta, -mg \\cos \\theta)$\"&gt;&lt;/span&gt;, sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; a massa do objeto e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img285.svg\" alt=\"$g$\"&gt;&lt;/span&gt; a aceleração gravitacional.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P421\" title=\"4P421\"&gt;\n                            A força de tensão com a haste é decomposta como &lt;!-- MATH\n                             $\\vec{t} = (0,T)$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img2105.svg\" alt=\"$\\vec{t} = (0,T)$\"&gt;&lt;/span&gt; sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\"&gt;&lt;/span&gt; o módulo da força de tensão na componente\n                            radial. A primeira coordenada é nula pois não há força de tensão com a haste no sentido tangencial.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P422\" title=\"4P422\"&gt;\n                            Como a velocidade do pêndulo é pequena, a força de atrito &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.76ex; vertical-align: -0.10ex; \" src=\"img/img2103.svg\" alt=\"$\\vec{r}$\"&gt;&lt;/span&gt; é considerada como sendo proporcional à velocidade\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img2.svg\" alt=\"$v$\"&gt;&lt;/span&gt;, isto é, &lt;!-- MATH\n                             $\\vec{r} = (-kv,0)$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2106.svg\" alt=\"$\\vec{r} = (-kv,0)$\"&gt;&lt;/span&gt;, para uma constante de proporcionalidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img635.svg\" alt=\"$k&gt;0$\"&gt;&lt;/span&gt;. O sinal negativo é consequência de que\n                            a força de atrito age no sentido contrário à velocidade. A segunda componente é nula pois a resistência não afeta o\n                            movimento radial. Na verdade, como a haste é considerada inextensível, não há movimento radial. A velocidade ainda deve\n                            ser dada em termos do deslocamento circular, isto é, &lt;!-- MATH\n                             $v = L \\frac{d\\theta}{dt}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2107.svg\" alt=\"$v = L \\frac{d\\theta}{dt}$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P423\" title=\"4P423\"&gt;\n                            Como vimos na seção anterior, &lt;!-- MATH\n                             $k = \\frac{\\rho A \\delta}{2}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.93ex; vertical-align: -0.83ex; \" src=\"img/img1992.svg\" alt=\"$k = \\frac{\\rho A \\delta}{2}$\"&gt;&lt;/span&gt;, em que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img1993.svg\" alt=\"$\\rho$\"&gt;&lt;/span&gt; é a densidade do ar, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img35.svg\" alt=\"$A$\"&gt;&lt;/span&gt; a área frontal do\n                            objeto e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1994.svg\" alt=\"$\\delta$\"&gt;&lt;/span&gt; um coeficiente que depende da forma do objeto. Para objetos esféricos considera-se &lt;!-- MATH\n                             $\\delta =\n                            \\frac{1}{2}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2108.svg\" alt=\"$\\delta =\n                            \\frac{1}{2}$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P424\" title=\"4P424\"&gt;\n                            De acordo com a segunda lei de Newton, temos &lt;!-- MATH\n                             $m\\vec{a} = \\vec{F}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img2109.svg\" alt=\"$m\\vec{a} = \\vec{F}$\"&gt;&lt;/span&gt;, sendo que &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img2110.svg\" alt=\"$\\vec{F}$\"&gt;&lt;/span&gt; é a força resultante do sistema.\n                            A aceleração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.76ex; vertical-align: -0.10ex; \" src=\"img/img2111.svg\" alt=\"$\\vec{a}$\"&gt;&lt;/span&gt; deve ser considerada somente na componente tangencial, em termos do deslocamento circular\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt;. Isto é, &lt;!-- MATH\n                             $\\vec{a} = \\left( \\frac{dv}{dt}, 0 \\right) = \\left( L \\frac{d^{2}\\theta}{dt^{2}}, 0 \\right)$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.98ex; vertical-align: -1.47ex; \" src=\"img/img2112.svg\" alt=\"$\\vec{a} = \\left( \\frac{dv}{dt}, 0 \\right) = \\left( L \\frac{d^{2}\\theta}{dt^{2}}, 0 \\right)$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P425\" title=\"4P425\"&gt;\n                            Assim, temos\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P426\" title=\"4P426\"&gt;&lt;table class=\"equation\"&gt;\n                            &lt;tbody&gt;&lt;tr&gt;\n                            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.45ex; vertical-align: -2.10ex; \" src=\"img/img2113.svg\" alt=\"$\\displaystyle \\left( mL \\frac{d^{2}\\theta}{dt^{2}}, 0 \\right)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.64ex; vertical-align: -0.50ex; \" src=\"img/img2114.svg\" alt=\"$\\displaystyle = m\\vec{a} = \\vec{F} = \\vec{p}+\\vec{t}+\\vec{r}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                            &lt;tr&gt;\n                            &lt;td&gt;&nbsp;&lt;/td&gt;\n                            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2115.svg\" alt=\"$\\displaystyle = (-mg {\\mathrm {sen}}\\theta, -mg \\cos \\theta) + (0,T) + (-kv,0)$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                            &lt;tr&gt;\n                            &lt;td&gt;&nbsp;&lt;/td&gt;\n                            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.82ex; vertical-align: -0.84ex; \" src=\"img/img2116.svg\" alt=\"$\\displaystyle = \\left( -mg {\\mathrm {sen}}\\theta - kL\\tfrac{d\\theta}{dt}, -mg \\cos \\theta + T \\right).$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                            \n                            \n                            &lt;p class=\" unidade\" id=\"4P427\" title=\"4P427\"&gt;\n                            Igualando cada uma das componentes temos que\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P428\" title=\"4P428\"&gt;&lt;table &gt;\n                            &lt;tbody&gt;&lt;tr&gt;\n                            &lt;td&gt;&nbsp;&lt;/td&gt;\n                            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.50ex; \" src=\"img/img2117.svg\" alt=\"$\\displaystyle mL\\theta'' + kL \\theta' + mg {\\mathrm {sen}}\\theta = 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                            &lt;a id=\"eqpend1\"&gt;&lt;/a&gt;(&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;16&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n                            &lt;tr&gt;\n                            &lt;td&gt;&nbsp;&lt;/td&gt;\n                            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img2118.svg\" alt=\"$\\displaystyle T = mg \\cos \\theta,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                            &lt;a id=\"eqpend2\"&gt;&lt;/a&gt;(&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;17&lt;/span&gt;)&lt;/td&gt;&lt;/tr&gt;\n                            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                            &lt;p class=\" unidade\" id=\"4P429\" style=\"text-indent: 0 !important;\" title=\"4P429\"&gt;\n                            sendo que &lt;!-- MATH\n                             $\\theta' = \\frac{d\\theta}{dt}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2119.svg\" alt=\"$\\theta' = \\frac{d\\theta}{dt}$\"&gt;&lt;/span&gt;. Este sistema está sujeito às condições iniciais em &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.88ex; vertical-align: -0.46ex; \" src=\"img/img1986.svg\" alt=\"$t_{0}$\"&gt;&lt;/span&gt;, dadas por\n                            &lt;!-- MATH\n                             $\\theta(t_{0}) = \\theta_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2120.svg\" alt=\"$\\theta(t_{0}) = \\theta_{0}$\"&gt;&lt;/span&gt; o ângulo inicial em que o pêndulo é solto e &lt;!-- MATH\n                             $\\theta'(t_{0}) = v_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2121.svg\" alt=\"$\\theta'(t_{0}) = v_{0}$\"&gt;&lt;/span&gt; a velocidade inicial\n                            do pêndulo. Se o pêndulo for solto do repouso, então naturalmente &lt;!-- MATH\n                             $\\theta'(t_{0}) = v_{0} = 0$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2122.svg\" alt=\"$\\theta'(t_{0}) = v_{0} = 0$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P430\" title=\"4P430\"&gt;\n                            A equação (&lt;a href=\"#eqpend1\"&gt;4.16&lt;/a&gt;) é uma equação diferencial não linear na variável &lt;!-- MATH\n                             $\\theta = \\theta(t)$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2101.svg\" alt=\"$\\theta = \\theta(t)$\"&gt;&lt;/span&gt;. Garantir a existência\n                            de uma solução pode ser complicado e mais complicado ainda talvez seja encontrar esta solução. Modificações podem ser\n                            feitas na equação (&lt;a href=\"#eqpend1\"&gt;4.16&lt;/a&gt;) a fim de facilitar a determinação de uma solução. Vamos estudar agora o caso em que a\n                            equação é liearizada, pois equações diferenciais lineares são mais fáceis de se obter solução. Vamos considerar então\n                            que o pêndulo oscile com variações pequenas do ângulo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt;. O limite (ver proposição &lt;a href=\"/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#limfundsin\"&gt;1.6&lt;/a&gt;),\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            \\lim_{\\theta \\to 0} \\frac{{\\mathrm {sen}}\\theta}{\\theta} = 1,\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P431\" title=\"4P431\"&gt;\n                            &lt;img style=\"height: 4.75ex; vertical-align: -1.78ex; \" src=\"img/img2123.svg\" alt=\"$\\displaystyle \\lim_{\\theta \\to 0} \\frac{{\\mathrm {sen}}\\theta}{\\theta} = 1, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P432\" style=\"text-indent: 0 !important;\" title=\"4P432\"&gt;\n                            sugere que para valores pequenos do argumento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt;, o numerador e o denominador são valores muito próximos. Podemos\n                            traduzir isto escrevendo &lt;!-- MATH\n                             ${\\mathrm {sen}}\\theta \\approx \\theta$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img2124.svg\" alt=\"${\\mathrm {sen}}\\theta \\approx \\theta$\"&gt;&lt;/span&gt; para valores pequenos de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt;. Para se ter uma ideia desta\n                            aproximação, o erro cometido ao aproximar &lt;!-- MATH\n                             ${\\mathrm {sen}}\\theta$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img2125.svg\" alt=\"${\\mathrm {sen}}\\theta$\"&gt;&lt;/span&gt; por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt; para um ângulo de &lt;!-- MATH\n                             $10^{\\circ}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.13ex; \" src=\"img/img2126.svg\" alt=\"$10^{\\circ}$\"&gt;&lt;/span&gt;, é menor que um\n                            milésimo.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P433\" title=\"4P433\"&gt;\n                            Nesta abordagem, o termo não linear &lt;!-- MATH\n                             ${\\mathrm {sen}}\\theta$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.68ex; vertical-align: -0.13ex; \" src=\"img/img2125.svg\" alt=\"${\\mathrm {sen}}\\theta$\"&gt;&lt;/span&gt; é substituído por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img1843.svg\" alt=\"$\\theta$\"&gt;&lt;/span&gt; e a equação (&lt;a href=\"#eqpend1\"&gt;4.16&lt;/a&gt;) torna-se\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P434\" title=\"4P434\"&gt;&lt;a id=\"eqpendlin\"&gt;&lt;/a&gt;&lt;!-- MATH\n                             \\begin{equation}\n                            mL\\theta'' + kL \\theta' + mg \\theta  = 0,\n                            \\end{equation}\n                             --&gt;\n                            &lt;table &gt;\n                            &lt;tbody&gt;&lt;tr&gt;\n                            &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.50ex; \" src=\"img/img2127.svg\" alt=\"$\\displaystyle mL\\theta'' + kL \\theta' + mg \\theta = 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                                (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;18&lt;/span&gt;) &lt;/td&gt;&lt;/tr&gt;\n                            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                            &lt;p class=\" unidade\" id=\"4P435\" style=\"text-indent: 0 !important;\" title=\"4P435\"&gt;\n                            que agora é uma equação diferencial linear. Esta equação diferencial possui soluções baseadas nas raízes da equação do\n                            segundo grau,\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            mLx^{2} + kLx + mg = 0,\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P436\" title=\"4P436\"&gt;\n                            &lt;img style=\"height: 2.53ex; vertical-align: -0.50ex; \" src=\"img/img2128.svg\" alt=\"$\\displaystyle mLx^{2} + kLx + mg = 0, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P437\" style=\"text-indent: 0 !important;\" title=\"4P437\"&gt;\n                            chamada de equação auxiliar. A respeito destas raízes, temos três casos a considerar.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P438\" title=\"4P438\"&gt;\n                            &lt;b&gt;Caso 1&lt;/b&gt;. Se &lt;!-- MATH\n                             $\\Delta = (k^{2}L^{2}-4m^{2}Lg) &gt; 0$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2129.svg\" alt=\"$\\Delta = (k^{2}L^{2}-4m^{2}Lg) &gt; 0$\"&gt;&lt;/span&gt; então a equação auxiliar possui duas raízes reais\n                            distintas\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            x_{1} = \\frac{-kL + \\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL}, \\qquad x_{2} = \\frac{-kL - \\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL},\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P439\" title=\"4P439\"&gt;\n                            &lt;img style=\"height: 5.14ex; vertical-align: -1.55ex; \" src=\"img/img2130.svg\" alt=\"$\\displaystyle x_{1} = \\frac{-kL + \\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL}, \\qquad x_{2} = \\frac{-kL - \\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL}, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P440\" style=\"text-indent: 0 !important;\" title=\"4P440\"&gt;\n                            e a solução de (&lt;a href=\"#eqpendlin\"&gt;4.18&lt;/a&gt;) é da forma\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            \\theta = \\theta(t) = C_{1}e^{x_{1}t} + C_{2}e^{x_{2}t},\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P441\" title=\"4P441\"&gt;\n                            &lt;img style=\"height: 2.56ex; vertical-align: -0.62ex; \" src=\"img/img2131.svg\" alt=\"$\\displaystyle \\theta = \\theta(t) = C_{1}e^{x_{1}t} + C_{2}e^{x_{2}t}, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P442\" style=\"text-indent: 0 !important;\" title=\"4P442\"&gt;\n                            para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; constantes a serem determinadas pelas condições iniciais &lt;!-- MATH\n                             $y(0) = \\theta_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2134.svg\" alt=\"$y(0) = \\theta_{0}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                             $y'(0) = v_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2135.svg\" alt=\"$y'(0) = v_{0}$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P443\" title=\"4P443\"&gt;\n                            Mas note que\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P444\" title=\"4P444\"&gt;&lt;table class=\"equation\"&gt;\n                            &lt;tbody&gt;&lt;tr&gt;\n                            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.51ex; vertical-align: -0.45ex; \" src=\"img/img2136.svg\" alt=\"$\\displaystyle x_{1}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.14ex; vertical-align: -1.55ex; \" src=\"img/img2137.svg\" alt=\"$\\displaystyle = \\frac{-kL + \\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL} &lt; \\frac{-kL + \\sqrt{k^{2}L^{2}}}{2mL} = 0$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                            &lt;tr&gt;\n                            &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img2138.svg\" alt=\"$\\displaystyle x_{2}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.14ex; vertical-align: -1.55ex; \" src=\"img/img2139.svg\" alt=\"$\\displaystyle = \\frac{-kL - \\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL} &lt; 0$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                            &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                            &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                            &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                            &lt;p class=\" unidade\" id=\"4P445\" style=\"text-indent: 0 !important;\" title=\"4P445\"&gt;\n                            e, portanto, o movimento angular do pêndulo decai a zero exponencialmente. Isto deve-se a um valor elevado da constante\n                            de proporcionalidade &lt;!-- MATH\n                             $k = \\frac{\\rho A \\delta}{2}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.93ex; vertical-align: -0.83ex; \" src=\"img/img1992.svg\" alt=\"$k = \\frac{\\rho A \\delta}{2}$\"&gt;&lt;/span&gt;. Valor alto o suficiente para tornar &lt;!-- MATH\n                             $k^{2}L^{2} &gt; 4m^{2}Lg$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.42ex; vertical-align: -0.50ex; \" src=\"img/img2140.svg\" alt=\"$k^{2}L^{2} &gt; 4m^{2}Lg$\"&gt;&lt;/span&gt;.\n                            Observemos que se a densidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img1993.svg\" alt=\"$\\rho$\"&gt;&lt;/span&gt;, do meio em que o pêndulo estiver imerso for alta, então esta situação é atingida.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P446\" title=\"4P446\"&gt;\n                            Neste caso, se o pêndulo passar pela solução de equilíbrio &lt;!-- MATH\n                             $\\theta(t) = 0$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2141.svg\" alt=\"$\\theta(t) = 0$\"&gt;&lt;/span&gt;, isto somente poderá ocorrer uma vez,\n                            exatamente no ponto\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            t = \\frac{mL}{\\sqrt{k^{2}L^{2}-4m^{2}Lg}} \\ln \\left( -\\frac{C_{2}}{C_{1}} \\right),\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P447\" title=\"4P447\"&gt;\n                            &lt;img style=\"height: 5.62ex; vertical-align: -2.48ex; \" src=\"img/img2142.svg\" alt=\"$\\displaystyle t = \\frac{mL}{\\sqrt{k^{2}L^{2}-4m^{2}Lg}} \\ln \\left( -\\frac{C_{2}}{C_{1}} \\right), $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P448\" style=\"text-indent: 0 !important;\" title=\"4P448\"&gt;\n                            e somente se &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; possuírem sinais contrários.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P449\" title=\"4P449\"&gt;\n                            &lt;b&gt;Caso 2&lt;/b&gt;. Se &lt;!-- MATH\n                             $\\Delta = (k^{2}L^{2}-4m^{2}Lg) = 0$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2143.svg\" alt=\"$\\Delta = (k^{2}L^{2}-4m^{2}Lg) = 0$\"&gt;&lt;/span&gt;, então a única raiz real da equação auxiliar é,\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            x = \\frac{-kL}{2mL} = \\frac{-k}{2m},\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P450\" title=\"4P450\"&gt;\n                            &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img2144.svg\" alt=\"$\\displaystyle x = \\frac{-kL}{2mL} = \\frac{-k}{2m}, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P451\" style=\"text-indent: 0 !important;\" title=\"4P451\"&gt;\n                            e, neste caso, a solução de (&lt;a href=\"#eqpendlin\"&gt;4.18&lt;/a&gt;) é dada por\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            \\theta = \\theta(t) = C_{1}e^{\\frac{-k}{2m}t} + C_{2}te^{\\frac{-k}{2m}t} = (C_{1} + C_{2}t) e^{\\frac{-k}{2m}t},\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P452\" title=\"4P452\"&gt;\n                            &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img2145.svg\" alt=\"$\\displaystyle \\theta = \\theta(t) = C_{1}e^{\\frac{-k}{2m}t} + C_{2}te^{\\frac{-k}{2m}t} = (C_{1} + C_{2}t) e^{\\frac{-k}{2m}t}, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P453\" title=\"4P453\"&gt;\n                            para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; constantes que satisfarão as condições iniciais &lt;!-- MATH\n                             $y(0) = \\theta_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2134.svg\" alt=\"$y(0) = \\theta_{0}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                             $y'(0) = v_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2135.svg\" alt=\"$y'(0) = v_{0}$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P454\" title=\"4P454\"&gt;\n                            Observe que ainda temos que a solução vai para zero quando &lt;!-- MATH\n                             $t \\to \\infty$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;. Também a solução passa uma única vez pela\n                            solução de equilíbrio, exatamente em\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            t = -\\frac{C_{1}}{C_{2}},\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P455\" title=\"4P455\"&gt;\n                            &lt;img style=\"height: 4.86ex; vertical-align: -1.89ex; \" src=\"img/img2146.svg\" alt=\"$\\displaystyle t = -\\frac{C_{1}}{C_{2}}, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P456\" style=\"text-indent: 0 !important;\" title=\"4P456\"&gt;\n                            também para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; com sinais contrários (pois &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img2147.svg\" alt=\"$t \\geq 0$\"&gt;&lt;/span&gt;).\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P457\" title=\"4P457\"&gt;\n                            &lt;b&gt;Caso 3&lt;/b&gt;. Se &lt;!-- MATH\n                             $\\Delta = (k^{2}L^{2}-4m^{2}Lg) &lt; 0$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2148.svg\" alt=\"$\\Delta = (k^{2}L^{2}-4m^{2}Lg) &lt; 0$\"&gt;&lt;/span&gt; então as raízes da equação auxiliar são os\n                            números complexos conjugados\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            x_{1} = \\overline{x_{2}} = \\frac{-k}{2m} + \\frac{\\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL} i,\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P458\" title=\"4P458\"&gt;\n                            &lt;img style=\"height: 5.14ex; vertical-align: -1.55ex; \" src=\"img/img2149.svg\" alt=\"$\\displaystyle x_{1} = \\overline{x_{2}} = \\frac{-k}{2m} + \\frac{\\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL} i, $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P459\" style=\"text-indent: 0 !important;\" title=\"4P459\"&gt;\n                            e então a solução da equação diferencial (&lt;a href=\"#eqpendlin\"&gt;4.18&lt;/a&gt;) toma a forma (que mais nos interessa)\n                            &lt;!-- MATH\n                             \\begin{displaymath}\n                            \\theta = \\theta(t) = e^{\\frac{-k}{2m}t} \\left( C_{1} \\cos(\\omega t) + C_{2} {\\mathrm {sen}}(\\omega t) \\right),\n                            \\end{displaymath}\n                             --&gt;\n                            &lt;/p&gt;\n                            &lt;div class=\"mathdisplay unidade\" id=\"4P460\" title=\"4P460\"&gt;\n                            &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img2150.svg\" alt=\"$\\displaystyle \\theta = \\theta(t) = e^{\\frac{-k}{2m}t} \\left( C_{1} \\cos(\\omega t) + C_{2} {\\mathrm {sen}}(\\omega t) \\right), $\"&gt;\n                            &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P461\" style=\"text-indent: 0 !important;\" title=\"4P461\"&gt;\n                            com &lt;!-- MATH\n                             $\\omega = \\frac{\\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.76ex; vertical-align: -0.83ex; \" src=\"img/img2151.svg\" alt=\"$\\omega = \\frac{\\sqrt{k^{2}L^{2}-4m^{2}Lg}}{2mL}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; constantes que satisfazem as condições\n                            iniciais &lt;!-- MATH\n                             $y(0) = \\theta_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2134.svg\" alt=\"$y(0) = \\theta_{0}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                             $y'(0) = v_{0}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2135.svg\" alt=\"$y'(0) = v_{0}$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P462\" title=\"4P462\"&gt;\n                            Observe que agora temos um movimento oscilatório. Mesmo assim, a presença da exponencial com potência negativa nos diz\n                            que o movimento tende a zero quando &lt;!-- MATH\n                             $t \\to \\infty$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;. Porém agora o valor da constante de proporcionalidade é pequeno.\n                            Para ser mais preciso, &lt;!-- MATH\n                             $k &lt; 2m\\sqrt{\\frac{g}{L}}$\n                             --&gt;\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.07ex; vertical-align: -1.44ex; \" src=\"img/img2152.svg\" alt=\"$k &lt; 2m\\sqrt{\\frac{g}{L}}$\"&gt;&lt;/span&gt;. Isto significa que a convergência para zero se dá de forma mais\n                            lenta, permitindo algum tempo de oscilação, antes do pêndulo parar. Esta parada ocorre na prática, mas teoricamente o\n                            pêndulo oscila para sempre com oscilação muito pequena.\n                            &lt;/p&gt;\n                            &lt;p class=\" unidade\" id=\"4P463\" title=\"4P463\"&gt;\n                            Para finalizar esta seção observe que a equação (&lt;a href=\"#eqpend2\"&gt;4.17&lt;/a&gt;) não foi utilizada até agora. Em verdade ela é útil para\n                            calcular a força de tração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img643.svg\" alt=\"$T$\"&gt;&lt;/span&gt;, exercida pelo fio sobre o objeto, depois que tivermos determinado uma expressão para\n                            &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2153.svg\" alt=\"$\\theta(t)$\"&gt;&lt;/span&gt;.\n                            &lt;/p&gt;",
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    "section": "4.7 Sistema massa-mola",
    "text": "4.7 Sistema massa-mola\n\n                                &lt;p class=\" unidade\" id=\"4P464\" title=\"4P464\"&gt;\n                                Vamos considerar que uma mola extensível, de comprimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2154.svg\" alt=\"$l$\"&gt;&lt;/span&gt; em repouso, esteja presa verticalmente a um suporte\n                                rígido. Prendemos então um objeto de massa &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; à extremidade livre da mola. Isto provocará uma distensão da mola, para\n                                um ponto de equilíbrio, por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\"&gt;&lt;/span&gt; unidades de comprimento.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P465\" title=\"4P465\"&gt;\n                                Parece natural que se deslocarmos a massa &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; e a soltarmos, esta massa oscilará em movimento de sobe e desce. Queremos\n                                um modelo para determinar a sua posição com o tempo. Para equacionar o problema, fixemos um sistema coordenado (só\n                                precisaremos da componente vertical) cuja origem está no ponto que dista &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2155.svg\" alt=\"$L=(l+s)$\"&gt;&lt;/span&gt; do suporte rígido e cresce no\n                                sentido do suporte.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P466\" title=\"4P466\"&gt;\n                                Designemos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1988.svg\" alt=\"$y(t)$\"&gt;&lt;/span&gt; a posição do objeto no instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;, ou mais precisamente, a posição da extremidade da mola no\n                                instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt;. Note então que a distância &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2156.svg\" alt=\"$d$\"&gt;&lt;/span&gt; entre o objeto e o suporte rígido, no instante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img98.svg\" alt=\"$t$\"&gt;&lt;/span&gt; é\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                d(t) = L - y(t).\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P467\" title=\"4P467\"&gt;\n                                &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2157.svg\" alt=\"$\\displaystyle d(t) = L - y(t). $\"&gt;\n                                &lt;/div&gt;\n                                \n                                &lt;p class=\" unidade\" id=\"4P468\" title=\"4P468\"&gt;\n                                De acordo com o nosso referencial duas forças unidimensionais atuam sobre o objeto. A força peso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img1837.svg\" alt=\"$\\vec{p}$\"&gt;&lt;/span&gt; e a força de\n                                tração da mola &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt;, ambas com mesma direção (vertical) e sentidos contrários.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P469\" title=\"4P469\"&gt;\n                                O peso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img1837.svg\" alt=\"$\\vec{p}$\"&gt;&lt;/span&gt;, considerado negativo por estar em sentido contrário ao eixo fixado, é dado por &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.82ex; vertical-align: -0.50ex; \" src=\"img/img1990.svg\" alt=\"$p = -mg$\"&gt;&lt;/span&gt;. A força de\n                                tração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt; é dada pela lei de Hooke. A lei de Hooke diz que a força de tração da mola é proporcional à distensão\n                                causada pela massa. Isto é, &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2158.svg\" alt=\"$t = ks$\"&gt;&lt;/span&gt; sendo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img635.svg\" alt=\"$k&gt;0$\"&gt;&lt;/span&gt; a constante de proporcionalidade, conhecida como constante de\n                                elasticidade da mola, que depende do material que a mola é composta.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P470\" title=\"4P470\"&gt;\n                                Agora note que, como o sistema está em equilíbrio, a força resultante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img2110.svg\" alt=\"$\\vec{F}$\"&gt;&lt;/span&gt; é nula. Isto é, &lt;!-- MATH\n                                 $\\vec{F} = \\vec{p} +\n                                \\vec{t} = \\vec{0}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.65ex; vertical-align: -0.50ex; \" src=\"img/img2159.svg\" alt=\"$\\vec{F} = \\vec{p} +\n                                \\vec{t} = \\vec{0}$\"&gt;&lt;/span&gt;, o que nos leva a\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                ks-mg = 0.\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P471\" title=\"4P471\"&gt;\n                                &lt;img style=\"height: 2.05ex; vertical-align: -0.50ex; \" src=\"img/img2160.svg\" alt=\"$\\displaystyle ks-mg = 0. $\"&gt;\n                                &lt;/div&gt;\n                                \n                                &lt;p class=\" unidade\" id=\"4P472\" title=\"4P472\"&gt;\n                                Desloquemos a massa por uma quantidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img2161.svg\" alt=\"$y_{0}$\"&gt;&lt;/span&gt; e deixamos o sistema livre para se movimentar. Agora a força de tensão\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.11ex; vertical-align: -0.10ex; \" src=\"img/img1838.svg\" alt=\"$\\vec{t}$\"&gt;&lt;/span&gt; depende também da posição &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2162.svg\" alt=\"$y = y(t)$\"&gt;&lt;/span&gt; do corpo. Temos assim,\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                \\vec{t} = k(s-y),\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P473\" title=\"4P473\"&gt;\n                                &lt;img style=\"height: 2.64ex; vertical-align: -0.62ex; \" src=\"img/img2163.svg\" alt=\"$\\displaystyle \\vec{t} = k(s-y), $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P474\" title=\"4P474\"&gt;\n                                pois a distenção da mola agora é &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2164.svg\" alt=\"$(s-y)$\"&gt;&lt;/span&gt;. De acordo com a segunda lei de Newton temos &lt;!-- MATH\n                                 $\\vec{F} = m\\vec{a}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img2165.svg\" alt=\"$\\vec{F} = m\\vec{a}$\"&gt;&lt;/span&gt;. Segue que\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                m\\vec{a} = \\vec{F} = \\vec{t} + \\vec{p} = ks - ky - mg = -ky,\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P475\" title=\"4P475\"&gt;\n                                &lt;img style=\"height: 2.64ex; vertical-align: -0.50ex; \" src=\"img/img2166.svg\" alt=\"$\\displaystyle m\\vec{a} = \\vec{F} = \\vec{t} + \\vec{p} = ks - ky - mg = -ky, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P476\" style=\"text-indent: 0 !important;\" title=\"4P476\"&gt;\n                                e, portanto, a equação que descreve o movimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1988.svg\" alt=\"$y(t)$\"&gt;&lt;/span&gt; do corpo é\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                m\\vec{a} = - ky.\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P477\" title=\"4P477\"&gt;\n                                &lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img2167.svg\" alt=\"$\\displaystyle m\\vec{a} = - ky. $\"&gt;\n                                &lt;/div&gt;\n                                \n                                &lt;p class=\" unidade\" id=\"4P478\" title=\"4P478\"&gt;\n                                Como a aceleração &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.76ex; vertical-align: -0.10ex; \" src=\"img/img2111.svg\" alt=\"$\\vec{a}$\"&gt;&lt;/span&gt; é a derivada segunda do movimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img1988.svg\" alt=\"$y(t)$\"&gt;&lt;/span&gt;, então a equação diferencial\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y'' + \\frac{k}{m} y = 0,\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P479\" title=\"4P479\"&gt;\n                                &lt;img style=\"height: 4.52ex; vertical-align: -1.55ex; \" src=\"img/img2168.svg\" alt=\"$\\displaystyle y'' + \\frac{k}{m} y = 0, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P480\" style=\"text-indent: 0 !important;\" title=\"4P480\"&gt;\n                                modela o movimento da massa &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt; com o passar do tempo. Ainda temos as condições iniciais\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y(0) = y_{0} \\quad \\text{e} \\quad y'(0) = v_{0}\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P481\" title=\"4P481\"&gt;\n                                &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2169.svg\" alt=\"$\\displaystyle y(0) = y_{0}$\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 2.44ex; vertical-align: -0.62ex; \" src=\"img/img2170.svg\" alt=\"$\\displaystyle \\quad y'(0) = v_{0} $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P482\" style=\"text-indent: 0 !important;\" title=\"4P482\"&gt;\n                                que siginificam respectivamente a posição inicial e a velocidade inicial (zero se o sistema é solto do repouso).\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P483\" title=\"4P483\"&gt;\n                                Temos então o Problema de Valor Inicial,\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P484\" title=\"4P484\"&gt;&lt;a id=\"PVIMaMo\"&gt;&lt;/a&gt;&lt;!-- MATH\n                                 \\begin{equation}\n                                \\left\\{ \\begin{array}{l} y'' + \\frac{k}{m} y = 0 \\\\y(0) = y_{0}, \\qquad y'(0) = v_{0} \\end{array} \\right.\n                                \\end{equation}\n                                 --&gt;\n                                &lt;table &gt;\n                                &lt;tbody&gt;&lt;tr&gt;\n                                &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img2171.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l} y'' + \\frac{k}{m} y = 0 \\\\ y(0) = y_{0}, \\qquad y'(0) = v_{0} \\end{array} \\right.$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                                &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: top;\"&gt;\n                                    (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;19&lt;/span&gt;)   &lt;/td&gt;&lt;/tr&gt;\n                                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                                \n                                \n                                &lt;p class=\" unidade\" id=\"4P485\" title=\"4P485\"&gt;\n                                Esta equação diferencial é uma equação linear homogênea de ordem 2. Podemos verificar que a função dada por\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y = C_{1} \\cos(\\omega t) + C_{2} {\\mathrm {sen}}(\\omega t),\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P486\" title=\"4P486\"&gt;\n                                &lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2172.svg\" alt=\"$\\displaystyle y = C_{1} \\cos(\\omega t) + C_{2} {\\mathrm {sen}}(\\omega t), $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P487\" style=\"text-indent: 0 !important;\" title=\"4P487\"&gt;\n                                com &lt;!-- MATH\n                                 $\\omega = \\sqrt{\\frac{k}{m}}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 4.07ex; vertical-align: -1.32ex; \" src=\"img/img2173.svg\" alt=\"$\\omega = \\sqrt{\\frac{k}{m}}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; são números reais quaisquer, é uma solução para a equação\n                                diferencial. Observe que o movimento é oscilatório em termos de senos e cossenos. As constantes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; podem\n                                ser determinadas substituindo-se as duas condições &lt;!-- MATH\n                                 $y(0) = y_{0}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2174.svg\" alt=\"$y(0) = y_{0}$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                                 $y'(0) = v_{0}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2135.svg\" alt=\"$y'(0) = v_{0}$\"&gt;&lt;/span&gt;. Temos assim,\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                \\left\\{ \\begin{array}{l} y_{0} = y(0) = C_{1} \\\\\n                                                            v_{0} = y'(0) = C_{2} \\omega \\end{array} \\right. \n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P488\" title=\"4P488\"&gt;\n                                &lt;img style=\"height: 6.51ex; vertical-align: -2.74ex; \" src=\"img/img2175.svg\" alt=\"$\\displaystyle \\left\\{ \\begin{array}{l} y_{0} = y(0) = C_{1} \\\\\n                                v_{0} = y'(0) = C_{2} \\omega \\end{array} \\right. $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P489\" style=\"text-indent: 0 !important;\" title=\"4P489\"&gt;\n                                donde a solução é\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y = y_{0} \\cos(\\omega t) + \\frac{v_{0}}{\\omega} {\\mathrm {sen}}(\\omega t).\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P490\" title=\"4P490\"&gt;\n                                &lt;img style=\"height: 4.03ex; vertical-align: -1.55ex; \" src=\"img/img2176.svg\" alt=\"$\\displaystyle y = y_{0} \\cos(\\omega t) + \\frac{v_{0}}{\\omega} {\\mathrm {sen}}(\\omega t). $\"&gt;\n                                &lt;/div&gt;\n                                \n                                &lt;p class=\" unidade\" id=\"4P491\" title=\"4P491\"&gt;\n                                Observe que para conhecer esta equação completamente ainda é necessário conhecer &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1840.svg\" alt=\"$\\omega$\"&gt;&lt;/span&gt; e, para isso, precisamos do\n                                valor da constante da mola &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\"&gt;&lt;/span&gt;. Este valor pode ser determinado medindo-se o deslocamento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img118.svg\" alt=\"$s$\"&gt;&lt;/span&gt; causado pela massa &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.16ex; vertical-align: -0.10ex; \" src=\"img/img1704.svg\" alt=\"$m$\"&gt;&lt;/span&gt;,\n                                pois como vimos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2177.svg\" alt=\"$(ks-mg)=0$\"&gt;&lt;/span&gt;, ou ainda, &lt;!-- MATH\n                                 $k = \\frac{mg}{s} = \\frac{p}{s}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.57ex; vertical-align: -0.83ex; \" src=\"img/img2178.svg\" alt=\"$k = \\frac{mg}{s} = \\frac{p}{s}$\"&gt;&lt;/span&gt;, onde &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.56ex; vertical-align: -0.50ex; \" src=\"img/img2077.svg\" alt=\"$p$\"&gt;&lt;/span&gt; é o peso do objeto (o módulo da força\n                                peso &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.15ex; vertical-align: -0.50ex; \" src=\"img/img1837.svg\" alt=\"$\\vec{p}$\"&gt;&lt;/span&gt;).\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P492\" title=\"4P492\"&gt;\n                                Este modelo pode ser complicado um pouco mais. Para ser mais preciso, este modelo é muito simples, pois supõe condições\n                                que na prática são irreais. As únicas forças consideradas são a força peso e a força de tração da mola e isto supõe a\n                                ausência de outras forças externas, como por exemplo, a resitência do ar. Este modelo precisa então de vácuo perfeito.\n                                Por este motivo, o sistema acima é dito sistema do movimento livre não amortecido.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P493\" title=\"4P493\"&gt;\n                                Um exemplo de complicação do problema é considerar que a mola “envelhece”. Em outras palavras, considerar que a\n                                constante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img50.svg\" alt=\"$k$\"&gt;&lt;/span&gt; da mola, seja variável com o tempo. Fisicamente isto significa que a mola perde suas propriedades\n                                iniciais de deformação com o passar do tempo.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P494\" title=\"4P494\"&gt;\n                                Poderíamos considerar a função de elasticidade da mola seja dada por &lt;!-- MATH\n                                 $ke^{-\\alpha t}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.92ex; vertical-align: -0.10ex; \" src=\"img/img2179.svg\" alt=\"$ke^{-\\alpha t}$\"&gt;&lt;/span&gt; com &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img635.svg\" alt=\"$k&gt;0$\"&gt;&lt;/span&gt; e &lt;!-- MATH\n                                 $\\alpha &gt; 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img638.svg\" alt=\"$\\alpha &gt; 0$\"&gt;&lt;/span&gt;. Temos\n                                então uma equação diferencial dada por &lt;!-- MATH\n                                 $y'' + \\frac{k}{m}e^{-\\alpha t}y = 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2180.svg\" alt=\"$y'' + \\frac{k}{m}e^{-\\alpha t}y = 0$\"&gt;&lt;/span&gt;. Outra função de elasticidade da mola que\n                                poderíamos considerar é &lt;!-- MATH\n                                 $k\\frac{1}{t}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2181.svg\" alt=\"$k\\frac{1}{t}$\"&gt;&lt;/span&gt; para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img635.svg\" alt=\"$k&gt;0$\"&gt;&lt;/span&gt; e então a equação diferencial se torna &lt;!-- MATH\n                                 $y'' + \\frac{k}{mt}y = 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2182.svg\" alt=\"$y'' + \\frac{k}{mt}y = 0$\"&gt;&lt;/span&gt;.\n                                Em ambos os casos temos equações diferenciais de ordem 2 com coeficientes variáveis e isto dificulta muito a obtenção\n                                de uma solução analítica.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P495\" title=\"4P495\"&gt;\n                                Outra complicação que podemos causar, que não dificulta determinar uma solução, é considerar que o corpo oscile imerso\n                                em algum fluido, como ar, água, óleo, entre outros. Isto obrigará a consideração de alguma força externa de atrito\n                                agindo sobre o sistema.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P496\" title=\"4P496\"&gt;\n                                Em geral, uma força de atrito é considerada como sendo proporcional a uma potência da velocidade. Como a velocidade de\n                                oscilação da massa é relativamente pequena, em geral a potência considerada é 1, isto é, a força de atrito &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.76ex; vertical-align: -0.10ex; \" src=\"img/img2103.svg\" alt=\"$\\vec{r}$\"&gt;&lt;/span&gt; é\n                                proporcional à velocidade &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.76ex; vertical-align: -0.10ex; \" src=\"img/img2183.svg\" alt=\"$\\vec{v}$\"&gt;&lt;/span&gt;.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P497\" title=\"4P497\"&gt;\n                                Nestes termos, consideremos\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                \\vec{r} = -\\lambda \\vec{v},\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P498\" title=\"4P498\"&gt;\n                                &lt;img style=\"height: 1.93ex; vertical-align: -0.27ex; \" src=\"img/img2184.svg\" alt=\"$\\displaystyle \\vec{r} = -\\lambda \\vec{v}, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P499\" style=\"text-indent: 0 !important;\" title=\"4P499\"&gt;\n                                para &lt;!-- MATH\n                                 $\\lambda &gt; 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1883.svg\" alt=\"$\\lambda &gt; 0$\"&gt;&lt;/span&gt;. O sinal negativo é decorrência de que a força de amortecimento é contrária à velocidade. Assim, a\n                                força resultante &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.25ex; vertical-align: -0.10ex; \" src=\"img/img2110.svg\" alt=\"$\\vec{F}$\"&gt;&lt;/span&gt; é\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                \\vec{F} = \\vec{p} + \\vec{t} + \\vec{r},\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P500\" title=\"4P500\"&gt;\n                                &lt;img style=\"height: 2.64ex; vertical-align: -0.50ex; \" src=\"img/img2185.svg\" alt=\"$\\displaystyle \\vec{F} = \\vec{p} + \\vec{t} + \\vec{r}, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P501\" style=\"text-indent: 0 !important;\" title=\"4P501\"&gt;\n                                e temos\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                m\\vec{a} = \\vec{F} = \\vec{t} + \\vec{p} + \\vec{r} = ks - ky - mg - \\lambda \\vec{v} = - ky - \\lambda \\vec{v}.\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P502\" title=\"4P502\"&gt;\n                                &lt;img style=\"height: 2.64ex; vertical-align: -0.50ex; \" src=\"img/img2186.svg\" alt=\"$\\displaystyle m\\vec{a} = \\vec{F} = \\vec{t} + \\vec{p} + \\vec{r} = ks - ky - mg - \\lambda \\vec{v} = - ky - \\lambda \\vec{v}. $\"&gt;\n                                &lt;/div&gt;\n                                \n                                &lt;p class=\" unidade\" id=\"4P503\" title=\"4P503\"&gt;\n                                Lembrando que a aceleração é a derivada segunda da posição (&lt;!-- MATH\n                                 $\\vec{a} = y''$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img2187.svg\" alt=\"$\\vec{a} = y''$\"&gt;&lt;/span&gt;) e que a velocidade é a derivada primeira\n                                da posição (&lt;!-- MATH\n                                 $\\vec{v} = y'$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.21ex; vertical-align: -0.50ex; \" src=\"img/img2188.svg\" alt=\"$\\vec{v} = y'$\"&gt;&lt;/span&gt;), então vem a equação diferencial\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P504\" title=\"4P504\"&gt;&lt;a id=\"eqmasmol\"&gt;&lt;/a&gt;&lt;!-- MATH\n                                 \\begin{equation}\n                                my'' + \\lambda y' + ky = 0,\n                                \\end{equation}\n                                 --&gt;\n                                &lt;table &gt;\n                                &lt;tbody&gt;&lt;tr&gt;\n                                &lt;td style=\"text-align:center;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.31ex; vertical-align: -0.50ex; \" src=\"img/img2189.svg\" alt=\"$\\displaystyle my'' + \\lambda y' + ky = 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                                &lt;td class=\"eqno\" style=\"text-align:right; vertical-align: middle;\"&gt;\n                                    (&lt;span class=\"arabic\"&gt;4&lt;/span&gt;.&lt;span class=\"arabic\"&gt;20&lt;/span&gt;)  &lt;/td&gt;&lt;/tr&gt;\n                                &lt;/tbody&gt;&lt;/table&gt; &lt;/div&gt;\n                                &lt;p class=\" unidade\" id=\"4P505\" style=\"text-indent: 0 !important;\" title=\"4P505\"&gt;\n                                sujeita às condições iniciais &lt;!-- MATH\n                                 $y(0) = y_{0}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.30ex; vertical-align: -0.62ex; \" src=\"img/img2174.svg\" alt=\"$y(0) = y_{0}$\"&gt;&lt;/span&gt; (posição inicial) e &lt;!-- MATH\n                                 $y'(0) = v_{0}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.34ex; vertical-align: -0.62ex; \" src=\"img/img2135.svg\" alt=\"$y'(0) = v_{0}$\"&gt;&lt;/span&gt; (velocidade inicial).\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P506\" title=\"4P506\"&gt;\n                                A solução desta equação agora depende agora das raízes da equação auxiliar quadrática,\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                mx^{2} + \\lambda x + k = 0,\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P507\" title=\"4P507\"&gt;\n                                &lt;img style=\"height: 2.30ex; vertical-align: -0.27ex; \" src=\"img/img2190.svg\" alt=\"$\\displaystyle mx^{2} + \\lambda x + k = 0, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P508\" style=\"text-indent: 0 !important;\" title=\"4P508\"&gt;\n                                e, portanto, do comportamento de &lt;!-- MATH\n                                 $\\Delta = \\lambda^{2} - 4mk$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img2191.svg\" alt=\"$\\Delta = \\lambda^{2} - 4mk$\"&gt;&lt;/span&gt;. Temos três casos a considerar.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P509\" title=\"4P509\"&gt;\n                                &lt;b&gt;Caso 1&lt;/b&gt;. Se &lt;!-- MATH\n                                 $\\Delta = (\\lambda^{2} - 4mk) &gt; 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2192.svg\" alt=\"$\\Delta = (\\lambda^{2} - 4mk) &gt; 0$\"&gt;&lt;/span&gt; então podemos verificar que a função dada por\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y(t) = C_{1} e^{x_{1} t} + C_{2} e^{x_{2} t},\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P510\" title=\"4P510\"&gt;\n                                &lt;img style=\"height: 2.56ex; vertical-align: -0.62ex; \" src=\"img/img2193.svg\" alt=\"$\\displaystyle y(t) = C_{1} e^{x_{1} t} + C_{2} e^{x_{2} t}, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P511\" style=\"text-indent: 0 !important;\" title=\"4P511\"&gt;\n                                com\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                x_{1} = \\frac{-\\lambda + \\sqrt{\\lambda^{2}-4mk}}{2m}, \\qquad \\text{e} \\qquad x_{2} = \\frac{-\\lambda - \\sqrt{\\lambda^{2}-4mk}}{2m},\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P512\" title=\"4P512\"&gt;\n                                &lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img2194.svg\" alt=\"$\\displaystyle x_{1} = \\frac{-\\lambda + \\sqrt{\\lambda^{2}-4mk}}{2m},$\"&gt;&nbsp; &nbsp;e&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img2195.svg\" alt=\"$\\displaystyle \\qquad x_{2} = \\frac{-\\lambda - \\sqrt{\\lambda^{2}-4mk}}{2m}, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P513\" style=\"text-indent: 0 !important;\" title=\"4P513\"&gt;\n                                é solução da equação diferencial (&lt;a href=\"#eqmasmol\"&gt;4.20&lt;/a&gt;) para quaisquer constantes reais &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt;.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P514\" title=\"4P514\"&gt;\n                                Observe que\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P515\" title=\"4P515\"&gt;&lt;table class=\"equation\"&gt;\n                                &lt;tbody&gt;&lt;tr&gt;\n                                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.51ex; vertical-align: -0.45ex; \" src=\"img/img2136.svg\" alt=\"$\\displaystyle x_{1}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.13ex; vertical-align: -1.55ex; \" src=\"img/img2196.svg\" alt=\"$\\displaystyle = \\frac{-\\lambda + \\sqrt{\\lambda^{2}-4mk}}{2m} &lt; \\frac{-\\lambda + \\sqrt{\\lambda^{2}}}{2m} = 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                                &lt;tr&gt;\n                                &lt;td style=\"text-align:right;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.50ex; vertical-align: -0.44ex; \" src=\"img/img2138.svg\" alt=\"$\\displaystyle x_{2}$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                                &lt;td style=\"text-align:left;\"&gt;&lt;span class=\"MATH\"&gt;&lt;img style=\"height: 5.04ex; vertical-align: -1.55ex; \" src=\"img/img2197.svg\" alt=\"$\\displaystyle = \\frac{-\\lambda - \\sqrt{\\lambda^{2}-4mk}}{2m} &lt; 0,$\"&gt;&lt;/span&gt;&lt;/td&gt;\n                                &lt;td class=\"eqno\" style=\"text-align:right\"&gt;\n                                &nbsp;&nbsp;&nbsp;&lt;/td&gt;&lt;/tr&gt;\n                                &lt;/tbody&gt;&lt;/table&gt;&lt;/div&gt;\n                                &lt;p class=\" unidade\" id=\"4P516\" style=\"text-indent: 0 !important;\" title=\"4P516\"&gt;\n                                e isto garante que, independentemente das constantes &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt;, ou da posição inicial e da velocidade inicial,\n                                a solução do sistema tende a zero exponencialmente quando &lt;!-- MATH\n                                 $t \\to \\infty$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;. De outra forma,\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                \\lim_{t \\to \\infty} y(t) = 0.\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P517\" title=\"4P517\"&gt;\n                                &lt;img style=\"height: 3.33ex; vertical-align: -1.66ex; \" src=\"img/img2198.svg\" alt=\"$\\displaystyle \\lim_{t \\to \\infty} y(t) = 0. $\"&gt;\n                                &lt;/div&gt;\n                                \n                                &lt;p class=\" unidade\" id=\"4P518\" title=\"4P518\"&gt;\n                                Isto significa que o movimento do corpo tende a cessar exponencialmente. É uma consequência imediata de uma constante\n                                de amortecimento &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2199.svg\" alt=\"$\\lambda$\"&gt;&lt;/span&gt; muito grande. Grande o suficiente para garantir que &lt;!-- MATH\n                                 $\\lambda^{2} - 4mk &gt; 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.20ex; vertical-align: -0.27ex; \" src=\"img/img2200.svg\" alt=\"$\\lambda^{2} - 4mk &gt; 0$\"&gt;&lt;/span&gt;. Neste caso\n                                dizemos que o sistema é super amortecido.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P519\" title=\"4P519\"&gt;\n                                &lt;b&gt;Caso 2&lt;/b&gt;. Se &lt;!-- MATH\n                                 $\\Delta = (\\lambda^{2}-4mk) = 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2201.svg\" alt=\"$\\Delta = (\\lambda^{2}-4mk) = 0$\"&gt;&lt;/span&gt; então podemos verificar que a função\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y(t) = C_{1} e^{x t} + C_{2}te^{x t},\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P520\" title=\"4P520\"&gt;\n                                &lt;img style=\"height: 2.56ex; vertical-align: -0.62ex; \" src=\"img/img2202.svg\" alt=\"$\\displaystyle y(t) = C_{1} e^{x t} + C_{2}te^{x t}, $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P521\" style=\"text-indent: 0 !important;\" title=\"4P521\"&gt;\n                                é solução da equação diferencial (&lt;a href=\"#eqmasmol\"&gt;4.20&lt;/a&gt;) para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; constantes reais e &lt;!-- MATH\n                                 $x = -\\frac{\\lambda}{2m}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.81ex; vertical-align: -0.83ex; \" src=\"img/img2203.svg\" alt=\"$x = -\\frac{\\lambda}{2m}$\"&gt;&lt;/span&gt;\n                                a única raiz real da equação auxiliar.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P522\" title=\"4P522\"&gt;\n                                Note que ainda temos &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.69ex; vertical-align: -0.14ex; \" src=\"img/img1317.svg\" alt=\"$x &lt; 0$\"&gt;&lt;/span&gt; e, portanto, a solução ainda decai (exponencialmente) para zero quando &lt;!-- MATH\n                                 $t \\to \\infty$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;. Este\n                                sistema é dito criticamente amortecido, pois ainda é amortecido, mas qualquer decréscimo na constante de amortecimento\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2199.svg\" alt=\"$\\lambda$\"&gt;&lt;/span&gt;, o movimento se tornará oscilatório.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P523\" title=\"4P523\"&gt;\n                                &lt;b&gt;Caso 3&lt;/b&gt;. Se &lt;!-- MATH\n                                 $\\Delta = (\\lambda^{2}-4mk) &lt; 0$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.55ex; vertical-align: -0.62ex; \" src=\"img/img2204.svg\" alt=\"$\\Delta = (\\lambda^{2}-4mk) &lt; 0$\"&gt;&lt;/span&gt; então temos que a função\n                                &lt;!-- MATH\n                                 \\begin{displaymath}\n                                y(t) = e^{\\frac{-\\lambda}{2m}t} \\left( C_{1}\\cos(\\omega t) + C_{2} {\\mathrm {sen}}(\\omega t) \\right),\n                                \\end{displaymath}\n                                 --&gt;\n                                &lt;/p&gt;\n                                &lt;div class=\"mathdisplay unidade\" id=\"4P524\" title=\"4P524\"&gt;\n                                &lt;img style=\"height: 3.01ex; vertical-align: -0.62ex; \" src=\"img/img2205.svg\" alt=\"$\\displaystyle y(t) = e^{\\frac{-\\lambda}{2m}t} \\left( C_{1}\\cos(\\omega t) + C_{2} {\\mathrm {sen}}(\\omega t) \\right), $\"&gt;\n                                &lt;/div&gt;&lt;p class=\" unidade\" id=\"4P525\" style=\"text-indent: 0 !important;\" title=\"4P525\"&gt;\n                                com &lt;!-- MATH\n                                 $\\omega = \\frac{\\sqrt{4mk-\\lambda^{2}}}{2m}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 3.22ex; vertical-align: -0.83ex; \" src=\"img/img2206.svg\" alt=\"$\\omega = \\frac{\\sqrt{4mk-\\lambda^{2}}}{2m}$\"&gt;&lt;/span&gt;, é solução da equação diferencial (&lt;a href=\"#eqmasmol\"&gt;4.20&lt;/a&gt;) para &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.00ex; vertical-align: -0.45ex; \" src=\"img/img2132.svg\" alt=\"$C_{1}$\"&gt;&lt;/span&gt; e\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.99ex; vertical-align: -0.44ex; \" src=\"img/img2133.svg\" alt=\"$C_{2}$\"&gt;&lt;/span&gt; constantes reais quaisquer. Observe que mesmo sendo um movimento oscilatório, o termo\n                                &lt;!-- MATH\n                                 $e^{\\frac{-\\lambda}{2m}t}$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 2.45ex; vertical-align: -0.10ex; \" src=\"img/img2207.svg\" alt=\"$e^{\\frac{-\\lambda}{2m}t}$\"&gt;&lt;/span&gt; tende a zero quando &lt;!-- MATH\n                                 $t \\to \\infty$\n                                 --&gt;\n                                &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.52ex; vertical-align: -0.10ex; \" src=\"img/img2072.svg\" alt=\"$t \\to \\infty$\"&gt;&lt;/span&gt;. Isto significa que este movimento oscilatório ainda\n                                tende a diminuir e cessar com o tempo. Porém isto ocorrerá de forma mais lenta permitindo algum tempo de oscilação do\n                                corpo antes da parada.\n                                &lt;/p&gt;\n                                &lt;p class=\" unidade\" id=\"4P526\" title=\"4P526\"&gt;\n                                Este tempo de oscilação naturalmente depende de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2199.svg\" alt=\"$\\lambda$\"&gt;&lt;/span&gt; que é a constante de proporcionalidade da força de atrito.\n                                Quanto menor o valor de &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.65ex; vertical-align: -0.10ex; \" src=\"img/img2199.svg\" alt=\"$\\lambda$\"&gt;&lt;/span&gt; mais tempo de oscilação antes de o corpo parar. Esta parada ocorre na prática, mas\n                                lembremos que teoricamente a oscilação ocorre para todo &lt;span class=\"MATH\"&gt;&lt;img style=\"height: 1.94ex; vertical-align: -0.38ex; \" src=\"img/img2147.svg\" alt=\"$t \\geq 0$\"&gt;&lt;/span&gt;.\n                                &lt;/p&gt;",
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    "text": "Eymard, Pierre; Lafon, Jean-Pierre. The number . Tradução (para o inglês) de Stephen S. Wilson, AMS,\n    Providence, Rhode Island, 2004.\n    \n    \n    \n    \n    Guidorizzi, Hamilton Luiz. Um curso de cálculo. Volume 4,  edição, Rio de Janeiro: LTC - Livros\n    técnicos e científicos, 2002.\n    \n    \n    \n    Iório, Valéria. EDP, um curso de graduação.  edição, Coleção Matemática Universitária. Rio de\n    Janeiro: Instituto de Matemática Pura e Aplicada, 2001.\n    \n    \n    \n    Leithold, Louis. O cálculo com geometria analítica. Vol 1.  edição. São Paulo: Editora Harbra, 1994.\n    \n    \n    \n    Lima, Elon L. Curso de análise. Vol 1. Projeto Euclides.  edição. Rio de Janeiro: Associação\n    Instituto Nacional de Matemática Pura e Aplicada, 2012.\n    \n    \n    \n    Sebah, Pascal; Gourdon, Xavier. Collection of series for . 2004.\n    \n     \n    \n    \n    Shervatov, V. G. Hyperbolic functions. Tradução (para o inglês) de A. Gordon Foster e Coley Mills, Jr.,\n    Tópicos em Matemática, Universidade de Chicago, 1963.\n    \n     \n    \n    \n    Swokowski, Earl William. Cálculo com geometria analítica. Vol 2.  edição. São Paulo: Makron Books,\n    1994.\n    \n     \n    \n    \n    Weisstein, Eric. MathWorld. Wolfram Research. http://mathworld.wolfram.com/ PiFormulas. Acesso em\n    08/10/2009.\n    \n     \n    \n    \n    Zill, Dennis G., Cullen, Michael R. Equações diferenciais. Vol 1.  edição. São Paulo: Makron Books,\n    2001.\n    \n     \n    \n    \n    Zill, Dennis G.; Shanahan, Patrick D. Curso introdutório à análise complexa com aplicações. Rio de Janeiro:\n    LTC, 2011.",
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    "text": "ângulo hiperbólico\n             ir \n        \n    \n        círculo trigonométrico\n             ir \n     circunferência trigonométrica\n     ir \n     cossecante\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     cossecante hiperbólica\n     ir \n    \n     derivada\n     ir \n    função\n         ir \n     função inversa\n     ir \n    \n    \n    \n     \n     cosseno\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     cosseno hiperbólico\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     cotangente\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     cotangente hiperbólica\n         ir\n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     \n    \n    \n     grado\n     ir \n     grau\n     ir \n    \n    \n    \n     hipérbole trigonométrica\n     ir \n    \n    \n    \n    \n     radiano\n     ir \n     relação fundamental\n    \n     da trigonometria circular\n     ir \n    \n    \n    \n    \n    \n     secante\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     secante hiperbólica\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     \n     função inversa\n     ir \n    \n     seno\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     seno hiperbólico\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     tangente\n     ir \n    \n     derivada\n     ir \n     função\n     ir \n     função inversa\n     ir \n    \n     tangente hiperbólica\n     ir \n    \n     derivada\n     ir \n    \n    \n     função\n     ir \n     função inversa\n     ir",
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