# Capítulo 2: Funções trigonométricas hiperbólicas {#SECTION00600000000000000000 .unnumbered}

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<div id="conteudo-capitulo">

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        <br />
        <br />
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        <p class=" unidade" id="2P1" title="2P1">
        Este capítulo é dedicado ao estudo das funções trigonométricas hiperbólicas. Iremos primeiramente estudar algumas
        propriedades importantes das hipérboles, para que possamos deduzir algumas relações envolvendo esta trigonometria.
        Vamos, depois, definir as seis funções trigonométricas hiperbólicas e um pequeno estudo sobre cada uma delas,
        principalmente no que diz respeito a derivada de tais funções. Feito isto, vamos estabelecer as funções trigonométricas
        hiperbólicas inversas e concluímos o capítulo com o estudo das derivadas das funções inversas.
        </p>

:::

##  2.1 Propriedades da hipérbole {#SECTION00610000000000000000}

::: {.raw_html}
            
            <p class=" unidade" id="2P2" title="2P2">
            Consideremos uma hipérbole de equação <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img634.svg" alt="$xy = k$" loading="lazy"></span>, para <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img635.svg" alt="$k>0$" loading="lazy"></span>. Para simplificar, vamos considerar que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img71.svg" alt="$y$" loading="lazy"></span> são ambos
            positivos, isto é, estamos tomando apenas um ramo da hipérbole. Os pontos desta curva são da forma <!-- MATH
             $(x, \frac{k}{x})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img636.svg" alt="$(x, \frac{k}{x})$" loading="lazy"></span>
            para <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img637.svg" alt="$x>0$" loading="lazy"></span> e o gráfico é a curva da figura <a href="#fighip">2.1</a>.
            </p>
            <a id="fighip"></a><a id="1702"></a>
            
            <div id="2I1" title="2I1" class="interativo unidade">
                <div class="controles_interatividade">
                
                <a href="interativo/fig2-1.html" target="_blank" class="btn_abrirInterativo">Ver maior</a><span class="barra"> | </span><span class="referencia" onclick="alert('A referência é: 2I1.')">Referência</span>
                
                </div>
                
                <iframe id="fig2-1" class="graficos" loading="lazy" src="interativo/fig2-1.html"></iframe>
                </div>
            
            <p class=" unidade" id="2P3" title="2P3">
            Dado um número real <!-- MATH
             $\alpha > 0$
             -->
            <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img638.svg" alt="$\alpha > 0$" loading="lazy"></span>, vamos considerar a transformação
            </p>
             <br>
            <div class="mathdisplay unidade" id="2P4" title="2P4"><a id="TransfT"></a><!-- MATH
             \begin{eqnarray}
            T : \mathbb{R}^{2} & \to & \mathbb{R}^{2} \nonumber \\
            (x,y) & \mapsto & T(x,y) = (\alpha x, \tfrac{1}{\alpha} y).
            \end{eqnarray}
             -->
            <table cellpadding="0" align="CENTER" width="100%">
            <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.13ex; vertical-align: -0.10ex; " src="img/img639.svg" alt="$\displaystyle T : \mathbb{R}^{2}$" loading="lazy"></td>
            <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
            <td align="LEFT" nowrap="" width="50%"><img style="height: 2.13ex; vertical-align: -0.10ex; " src="img/img640.svg" alt="$\displaystyle \mathbb{R}^{2}$" loading="lazy"></td>
            <td class="eqno" width="10" align="RIGHT">
            &nbsp;</td></tr>
            <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img641.svg" alt="$\displaystyle (x,y)$" loading="lazy"></td>
            <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
            <td align="LEFT" nowrap="" width="50%"><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"></td>
            <td class="eqno" width="10" align="RIGHT">
            (<span class="arabic">2</span>.<span class="arabic">1</span>)</td></tr>
            </tbody></table></div>
            <br clear="ALL">
            
            <p class=" unidade" id="2P5" title="2P5">
            Esta transformação é conhecida como deslocamento ou deslizamento sobre a hipérbole. Isto se deve ao fato de que
            <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> leva pontos da hipérbole na hipérbole (ver proposição <a href="#prophip">2.1</a>). Esta transformação é bastante importante no
            nosso estudo e possui propriedades interessantes. As próximas proposições evidenciam algumas destas propriedades.
            Outras propriedades podem ser encontradas em [<a href="/trigonometria-hiperbolica/referencias#Shervatov">7</a>, Shervatov].
            
            </p>
            
            <div id="2Teo1" title="2Teo1" class=" unidade"><a id="prophip"><b>Proposição  <span class="arabic">2</span>.<span class="arabic">1</span></b></a> 
            <i>A transformação <!-- MATH
             $T: \mathbb{R}^{2} \to \mathbb{R}^{2}$
             -->
            <span class="MATH"><img style="height: 2.02ex; vertical-align: -0.10ex; " src="img/img644.svg" alt="$T: \mathbb{R}^{2} \to \mathbb{R}^{2}$" loading="lazy"></span>, definida por </i>(<a href="#TransfT">2.1</a>)<i>, possui as seguintes propriedades: 
            <br>
            <!-- MATH
             $\mathbf{(a)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img645.svg" alt="$\mathbf{(a)}$" loading="lazy"></span> <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> leva pontos da hipérbole em pontos da hipérbole. 
            <br>
            <!-- MATH
             $\mathbf{(b)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img646.svg" alt="$\mathbf{(b)}$" loading="lazy"></span> <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> leva retas do plano em retas. 
            <br>
            <!-- MATH
             $\mathbf{(c)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img647.svg" alt="$\mathbf{(c)}$" loading="lazy"></span> <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> preserva a razão entre os comprimentos de segmentos de uma mesma reta. 
            <br>
            <!-- MATH
             $\mathbf{(d)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img648.svg" alt="$\mathbf{(d)}$" loading="lazy"></span> <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> leva retas paralelas em retas paralelas. 
            <br>
            <!-- MATH
             $\mathbf{(e)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img649.svg" alt="$\mathbf{(e)}$" loading="lazy"></span> <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> preserva as assíntotas da hipérbole.
            </i></div>
            <div><i>Prova</i>.
            
            Dado <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img650.svg" alt="$a > 0$" loading="lazy"></span>, o ponto <!-- MATH
             $A = (a,\frac{k}{a})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img651.svg" alt="$A = (a,\frac{k}{a})$" loading="lazy"></span> pertencente à hipérbole e temos
            <!-- MATH
             \begin{displaymath}
            T(A) = T(a, \tfrac{k}{a}) = (\alpha a, \tfrac{1}{\alpha} \tfrac{k}{a}) = (\alpha a, \tfrac{k}{\alpha a}),
            \end{displaymath}
             -->
            
            <div class="mathdisplay unidade" id="2P6" title="2P6">
            <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}), $" loading="lazy">
            </div>
            sendo que claramente o membro da direita é um ponto que ainda está sobre a hipérbole, uma vez que satisfaz a equação
            <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img634.svg" alt="$xy = k$" loading="lazy"></span>. Isto prova <!-- MATH
             $\mathbf{(a)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img645.svg" alt="$\mathbf{(a)}$" loading="lazy"></span>. Para provar <!-- MATH
             $\mathbf{(b)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img646.svg" alt="$\mathbf{(b)}$" loading="lazy"></span> consideremos a parametrização <!-- MATH
             $(x,y) = (c,d) +
            t(a,b) = (at+c,bt+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img653.svg" alt="$(x,y) = (c,d) +
            t(a,b) = (at+c,bt+d)$" loading="lazy"></span> que descreve os pontos de uma reta, fazendo <span class="MATH"><img style="height: 1.52ex; vertical-align: -0.10ex; " src="img/img98.svg" alt="$t$" loading="lazy"></span> variar em <!-- MATH
             $\mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>, para quaisquer <!-- MATH
             $a,b,c,d \in \mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img654.svg" alt="$a,b,c,d \in \mathbb{R}$" loading="lazy"></span>,
            desde que <!-- MATH
             $(a,b) \neq (0,0)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img655.svg" alt="$(a,b) \neq (0,0)$" loading="lazy"></span> pois é o vetor diretor da reta. Então aplicando <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span>,
            <!-- MATH
             \begin{displaymath}
            T(at+c,bt+d) = (\alpha at + \alpha c, \tfrac{bt+d}{\alpha}) = ( \alpha at + \alpha c, \tfrac{b}{\alpha} t + \tfrac{d}{\alpha}),
            \end{displaymath}
             -->
            
            <div class="mathdisplay unidade" id="2P7" title="2P7">
            <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}), $" loading="lazy">
            </div>
            e obviamente os pontos do membro da direita descrevem uma reta fazendo <span class="MATH"><img style="height: 1.52ex; vertical-align: -0.10ex; " src="img/img98.svg" alt="$t$" loading="lazy"></span> variar em <!-- MATH
             $\mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>. Para o item <!-- MATH
             $\mathbf{(c)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img647.svg" alt="$\mathbf{(c)}$" loading="lazy"></span>,
            consideremos novamente a parametrização <!-- MATH
             $(at+c,bt+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img657.svg" alt="$(at+c,bt+d)$" loading="lazy"></span> de uma reta qualquer. O comprimento de um segmento
            <!-- MATH
             $\overline{AB}$
             -->
            <span class="MATH"><img style="height: 2.07ex; vertical-align: -0.10ex; " src="img/img658.svg" alt="$\overline{AB}$" loading="lazy"></span> é a distância entre <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span>. Se <!-- MATH
             $A = (at_{1}+c, bt_{1}+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img659.svg" alt="$A = (at_{1}+c, bt_{1}+d)$" loading="lazy"></span> e <!-- MATH
             $B = (at_{2}+c, bt_{2}+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img660.svg" alt="$B = (at_{2}+c, bt_{2}+d)$" loading="lazy"></span>, então
            
            <div class="mathdisplay unidade" id="2P8" title="2P8"><table class="equation">
            <tbody><tr>
            <td style="text-align:right;"><span class="MATH"><img style="height: 2.07ex; vertical-align: -0.10ex; " src="img/img661.svg" alt="$\displaystyle \overline{AB}$" loading="lazy"></span></td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            <tr>
            <td>&nbsp;</td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            <tr>
            <td>&nbsp;</td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            </tbody></table></div>
            
            
            <p class=" unidade" id="2P9" title="2P9">
            Se <!-- MATH
             $\overline{CD}$
             -->
            
            <span class="MATH"><img style="height: 2.07ex; vertical-align: -0.10ex; " src="img/img665.svg" alt="$\overline{CD}$" loading="lazy"></span> é outro segmento desta reta com <!-- MATH
             $C = (at_{3}+c, bt_{3}+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img666.svg" alt="$C = (at_{3}+c, bt_{3}+d)$" loading="lazy"></span> e <!-- MATH
             $D = (at_{4}+c, bt_{4}+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img667.svg" alt="$D = (at_{4}+c, bt_{4}+d)$" loading="lazy"></span>, então da
            mesma forma, o comprimento do segmento <!-- MATH
             $\overline{CD}$
             -->
            <span class="MATH"><img style="height: 2.07ex; vertical-align: -0.10ex; " src="img/img665.svg" alt="$\overline{CD}$" loading="lazy"></span> é igual a <!-- MATH
             $|t_{4}-t_{3}|\sqrt{a^{2} + b^{2}}$
             -->
            <span class="MATH"><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"></span> e a razão entre os
            segmentos é
            <!-- MATH
             \begin{displaymath}
            \frac{\overline{CD}}{\overline{AB}}
             = \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}|}, 
            \end{displaymath}
             -->
            
            <div class="mathdisplay unidade" id="2P10" title="2P10">
            <img style="height: 5.87ex; vertical-align: -2.38ex; " src="img/img669.svg" alt="$\displaystyle \frac{\overline{CD}}{\overline{AB}}
            = \frac{\vert t_{4}-t_{3}\v...
            ...\sqrt{a^{2} + b^{2}}} = \frac{\vert t_{4}-t_{3}\vert}{\vert t_{2}-t_{1}\vert}, $" loading="lazy">
            </div>
            pois <!-- MATH
             $\sqrt{a^{2}+b^{2}} = 0$
             -->
            <span class="MATH"><img style="height: 2.38ex; vertical-align: -0.31ex; " src="img/img670.svg" alt="$\sqrt{a^{2}+b^{2}} = 0$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P11" title="2P11">
            Vamos então provar que se <span class="MATH"><img style="height: 2.34ex; vertical-align: -0.62ex; " src="img/img671.svg" alt="$A' = T(A)$" loading="lazy"></span>, <span class="MATH"><img style="height: 2.34ex; vertical-align: -0.62ex; " src="img/img672.svg" alt="$B' = T(B)$" loading="lazy"></span>, <span class="MATH"><img style="height: 2.34ex; vertical-align: -0.62ex; " src="img/img673.svg" alt="$C' = T(C)$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.34ex; vertical-align: -0.62ex; " src="img/img674.svg" alt="$D' = T(D)$" loading="lazy"></span>, então
            <!-- MATH
             $\frac{\overline{C'D'}}{\overline{A'B'}} = \frac{|t_{4}-t_{3}|}{|t_{2}-t_{1}|}$
             -->
            <span class="MATH"><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"></span>. De fato, <!-- MATH
             $A' = (\alpha at_{1} + \alpha
            c, \frac{b}{\alpha}t_{1} + \frac{d}{\alpha})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img676.svg" alt="$A' = (\alpha at_{1} + \alpha
            c, \frac{b}{\alpha}t_{1} + \frac{d}{\alpha})$" loading="lazy"></span>, <!-- MATH
             $B' = (\alpha at_{2} + \alpha c, \frac{b}{\alpha}t_{2} +
            \frac{d}{\alpha})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img677.svg" alt="$B' = (\alpha at_{2} + \alpha c, \frac{b}{\alpha}t_{2} +
            \frac{d}{\alpha})$" loading="lazy"></span>, <!-- MATH
             $C' = (\alpha at_{3} + \alpha c, \frac{b}{\alpha}t_{3} + \frac{d}{\alpha})$
             -->
            <span class="MATH"><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"></span> e <!-- MATH
             $D' = (\alpha at_{4}
            + \alpha c, \frac{b}{\alpha}t_{4} + \frac{d}{\alpha})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img679.svg" alt="$D' = (\alpha at_{4}
            + \alpha c, \frac{b}{\alpha}t_{4} + \frac{d}{\alpha})$" loading="lazy"></span>. Então
            </p>
            <div class="mathdisplay unidade" id="2P12" title="2P12"><table class="equation">
            <tbody><tr>
            <td style="text-align:right;"><span class="MATH"><img style="height: 2.08ex; vertical-align: -0.10ex; " src="img/img680.svg" alt="$\displaystyle \overline{A'B'}$" loading="lazy"></span></td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            <tr>
            <td>&nbsp;</td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            <tr>
            <td>&nbsp;</td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            </tbody></table></div>
            <p style="text-indent:0px !important;" class=" unidade" id="2P13" title="2P13">
            e da mesma forma
            </p><!-- MATH
             \begin{displaymath}
            \overline{C'D'} = |t_{4}-t_{3}|\sqrt{\alpha^{2}a^{2} + \frac{b^{2}}{\alpha^{2}}},
            \end{displaymath}
             -->
            
            <div class="mathdisplay unidade" id="2P14" title="2P14">
            <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">
            </div><p style="text-indent:0px !important;" class=" unidade" id="2P15" title="2P15">
            e, portanto,
            </p><!-- MATH
             \begin{displaymath}
            \frac{\overline{C'D'}}{\overline{A'B'}}
             = \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}|}
             = \frac{\overline{CD}}{\overline{AB}}, 
            \end{displaymath}
             -->
            
            <div class="mathdisplay unidade" id="2P16" title="2P16">
            <img style="height: 8.53ex; vertical-align: -3.75ex; " src="img/img685.svg" alt="$\displaystyle \frac{\overline{C'D'}}{\overline{A'B'}}
            = \frac{\vert t_{4}-t_{...
            ...}-t_{3}\vert}{\vert t_{2}-t_{1}\vert}
            = \frac{\overline{CD}}{\overline{AB}}, $" loading="lazy">
            </div>
            <p style="text-indent:0px !important;" class=" unidade" id="2P17" title="2P17">e isso prova o item <!-- MATH
             $\mathbf{(c)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img647.svg" alt="$\mathbf{(c)}$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P18" title="2P18">
            Para provar <!-- MATH
             $\mathbf{(d)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img648.svg" alt="$\mathbf{(d)}$" loading="lazy"></span>, tomemos duas retas paralelas <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span>, isto é, os coeficientes são proporcionais. Tomemos
            as parametrizações <!-- MATH
             $r:(at+c,bt+d)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img686.svg" alt="$r:(at+c,bt+d)$" loading="lazy"></span> e <!-- MATH
             $s:( (ma)t + e, (mb)t + f)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img687.svg" alt="$s:( (ma)t + e, (mb)t + f)$" loading="lazy"></span>, para <!-- MATH
             $t \in \mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img688.svg" alt="$t \in \mathbb{R}$" loading="lazy"></span>. Então, <!-- MATH
             $T(r) = ( (\alpha a) t +
            \alpha c, \frac{b}{\alpha}t + \frac{d}{\alpha})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img689.svg" alt="$T(r) = ( (\alpha a) t +
            \alpha c, \frac{b}{\alpha}t + \frac{d}{\alpha})$" loading="lazy"></span> e <!-- MATH
             $T(s) = ( (\alpha ma) t + \alpha e, \frac{mb}{\alpha}t +
            \frac{f}{\alpha})$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img690.svg" alt="$T(s) = ( (\alpha ma) t + \alpha e, \frac{mb}{\alpha}t +
            \frac{f}{\alpha})$" loading="lazy"></span> e, claramente, <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img691.svg" alt="$T(r)$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img692.svg" alt="$T(s)$" loading="lazy"></span> são paralelas, já que os seus coeficientes são proporcionais.
            </p>
            <p>
            Finalmente vamos a <!-- MATH
             $\mathbf{(e)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img649.svg" alt="$\mathbf{(e)}$" loading="lazy"></span>. As assíntotas da hipérbole são os eixos coordenados. As suas parametrizações são
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img693.svg" alt="$(t,0)$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img694.svg" alt="$(0,t)$" loading="lazy"></span>. Obviamente <!-- MATH
             $T(t,0) = (\alpha t,0)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img695.svg" alt="$T(t,0) = (\alpha t,0)$" loading="lazy"></span>, que continua sendo o eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span> e <!-- MATH
             $T(0,t) = (0, \frac{1}{\alpha}
            t)$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img696.svg" alt="$T(0,t) = (0, \frac{1}{\alpha}
            t)$" loading="lazy"></span>, que continua sendo o eixo <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img71.svg" alt="$y$" loading="lazy"></span>.
            <span style="float: right"><img style="height: 1.59ex; vertical-align: -0.10ex; " src="img/img193.svg" alt="$\qedsymbol$" loading="lazy"></span>
            </p>
            </div>
        
            <p class=" unidade" id="2P19" title="2P19">
            <b>Nota</b>: Observe que, na transformação dada em (<a href="#TransfT">2.1</a>), para <!-- MATH
             $\alpha = 1$
             -->
            <span class="MATH"><img style="height: 1.66ex; vertical-align: -0.11ex; " src="img/img697.svg" alt="$\alpha = 1$" loading="lazy"></span> temos a aplicação identidade. Para <!-- MATH
             $\alpha >
            1$
             -->
            <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img698.svg" alt="$\alpha >
            1$" loading="lazy"></span>, um dado ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> será deslocado por <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> no sentido do crescimento do eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>. E se <!-- MATH
             $0 < \alpha < 1$
             -->
            <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img699.svg" alt="$0 < \alpha < 1$" loading="lazy"></span>, então o
            deslocamento de um determinado ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> da hipérbole, se dará no sentido contrário ao do crescimento do eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>.
            <!-- MATH
             $\blacksquare$
             -->
            <span style="float: right" class="MATH"><img style="height: 1.61ex; vertical-align: -0.10ex; " src="img/img279.svg" alt="$\blacksquare$" loading="lazy"></span>
             </p>
            <br>
           
            <div id="2Teo2" title="2Teo2" class=" unidade"><a id="nochangearea"><b>Proposição  <span class="arabic">2</span>.<span class="arabic">2</span></b></a> &nbsp; 
            <i>A transformação <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> não altera a área de figuras do plano.
            </i></div>
            
          
            <div><i>Prova</i>.
            
            De fato, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> pode ser escrita como composição das transformações
            <!-- MATH
             \begin{displaymath}
            \begin{array}{rcl}
            T_{1} : \mathbb{R}^{2} & \to & \mathbb{R}^{2} \\
            (x,y) & \mapsto & (\alpha x, y)
            \end{array} \qquad \qquad \text{e} \qquad \qquad
            \begin{array}{rcl}
            T_{2} : \mathbb{R}^{2} & \to & \mathbb{R}^{2} \\
            (x,y) & \mapsto & (x, \tfrac{1}{\alpha}y).
            \end{array}
            \end{displaymath}
             -->
            
            <div class="mathdisplay unidade" id="2P20" title="2P20">
            <img style="height: 6.47ex; vertical-align: -2.72ex; " src="img/img700.svg" alt="$\displaystyle \begin{array}{rcl}
            T_{1} : \mathbb{R}^{2} &amp; \to &amp; \mathbb{R}^{2} \\
            (x,y) &amp; \mapsto &amp; (\alpha x, y)
            \end{array}$" loading="lazy">&nbsp; &nbsp;e<img src="img/img701.svg" alt="\begin{displaymath}\qquad \qquad
            \begin{array}{rcl}
            T_{2} : \mathbb{R}^{2} &amp; \...
            ...2} \\
            (x,y) &amp; \mapsto &amp; (x, \tfrac{1}{\alpha}y).
            \end{array}\end{displaymath}" loading="lazy">
            </div>
            
            <p class=" unidade" id="2P21" title="2P21">
            Sabemos da geometria plana que a transformação <span class="MATH"><img style="height: 2.00ex; vertical-align: -0.45ex; " src="img/img702.svg" alt="$T_{1}$" loading="lazy"></span> altera a área de uma figura plana multiplicando esta área por
            <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img13.svg" alt="$\alpha$" loading="lazy"></span> e a segunda transformação multiplica a área de uma figura por <!-- MATH
             $\frac{1}{\alpha}$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img703.svg" alt="$\frac{1}{\alpha}$" loading="lazy"></span>. A composta das duas
            aplicações então multiplica a área de figuras primeiro por <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img13.svg" alt="$\alpha$" loading="lazy"></span> e depois por <!-- MATH
             $\frac{1}{\alpha}$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img703.svg" alt="$\frac{1}{\alpha}$" loading="lazy"></span> e, portanto, não
            altera a área de figuras planas.
            <span style="float: right"><img style="height: 1.59ex; vertical-align: -0.10ex; " src="img/img193.svg" alt="$\qedsymbol$" loading="lazy"></span>
            </p></div><p class=" unidade" id="2P22" title="2P22"></p>
            
            <p class=" unidade" id="2P23" title="2P23">
            A figura <a href="#fighiptrig">2.2</a> representa um braço da hipérbole de equação cartesiana <!-- MATH
             $x^{2} - y^{2} = 1$
             -->
            <span class="MATH"><img style="height: 2.42ex; vertical-align: -0.50ex; " src="img/img704.svg" alt="$x^{2} - y^{2} = 1$" loading="lazy"></span>. Esta “metade”
            de hipérbole é conhecida como hipérbole trigonométrica. <a name="1922"></a>
            </p>
            <p class=" unidade" id="2P24" title="2P24">
            Note que, se rotacionarmos o gráfico da figura <a href="#fighiptrig">2.2</a>, <!-- MATH
             $45^{\circ}$
             -->
            <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.13ex; " src="img/img705.svg" alt="$45^{\circ}$" loading="lazy"></span> no sentido anti-horário com relação a
            origem, este braço de hipérbole se torna o braço de hipérbole da figura <a href="#fighip">2.1</a>, bastando apenas ajustar o valor
            de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img50.svg" alt="$k$" loading="lazy"></span>. Também as assíntotas <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img706.svg" alt="$y = \pm x$" loading="lazy"></span>, após esta rotação, se tornam os eixos coordenados da figura <a href="#fighip">2.1</a>.
            Isso significa que, por uma rotação, as propriedades listadas nas Proposições <a href="#prophip">2.1</a> e <a href="#nochangearea">2.2</a> são
            válidas também na hipérbole trigonométrica e suas assíntotas <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img706.svg" alt="$y = \pm x$" loading="lazy"></span>. Isto porque a rotação (de <!-- MATH
             $45^{\circ}$
             -->
            <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.13ex; " src="img/img705.svg" alt="$45^{\circ}$" loading="lazy"></span>), é
            um movimento rígido e preserva comprimento de segmentos, relação de paralelismo e medidas de áreas.
            </p>
            <div class="CENTER"><a id="fighiptrig"></a><a id="1932"></a>
            <table id="2I2" title="2I2">
            <caption class="BOTTOM"><strong>Figura 2.2:</strong>
            Hipérbole trigonométrica.</caption>
            <tbody><tr><td>
            <div class="CENTER">
            <img src="img/hiptrig.png" alt="Image hiptrig" loading="lazy"> </div></td></tr>
            </tbody></table>
            </div>
            
            <p class=" unidade" id="2P25" title="2P25">
            Chamemos <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img34.svg" alt="$V=(1,0)$" loading="lazy"></span> o vértice da hipérbole da figura <a href="#fighiptrig">2.2</a>. Consideremos um ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> sobre a hipérbole
            situado no primeiro quadrante (fig. <a href="#figtrig1">2.3</a>). Pelo ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> traçamos a perpendicular <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img68.svg" alt="$AP$" loading="lazy"></span> ao eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>. Marcamos
            o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span> simétrico de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> com relação ao eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>. O segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img707.svg" alt="$AB$" loading="lazy"></span> é dito segmento conjugado do segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span>, pois
            o prolongamento de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span> encontra <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img707.svg" alt="$AB$" loading="lazy"></span> no ponto médio <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img4.svg" alt="$P$" loading="lazy"></span> de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img707.svg" alt="$AB$" loading="lazy"></span>. O ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span> está então sobre a hipérbole no quarto
            quadrante e o segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img707.svg" alt="$AB$" loading="lazy"></span> é também perpendicular ao eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>. Traçamos pelo ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> a reta <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> paralela a assíntota
            <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img708.svg" alt="$y=x$" loading="lazy"></span> e pelo ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span> a reta <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span> paralela a assíntota <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img709.svg" alt="$y=-x$" loading="lazy"></span>. As retas <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span> se encontram no ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img101.svg" alt="$M$" loading="lazy"></span>, sobre o
            eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>, formando o triângulo <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img710.svg" alt="$AMB$" loading="lazy"></span> retângulo em <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img101.svg" alt="$M$" loading="lazy"></span>.
            </p>
            <div class="CENTER"><a id="figtrig1"></a><a id="1939"></a>
            <table id="2I3" title="2I3">
            <caption class="BOTTOM"><strong>Figura 2.3:</strong>
             Construindo propriedades adicionais na hipérbole. </caption>
            <tbody><tr><td>
            <div class="CENTER">
            <img src="img/fighip01.png" alt="Image fighip01" loading="lazy"> </div></td></tr>
            </tbody></table>
            </div>
            
            <p class=" unidade" id="2P26" title="2P26">
            Aplicamos agora um deslizamento sobre a hipérbole, isto é, aplicamos a transformação <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> dada em (<a href="#TransfT">2.1</a>).
            Obtemos assim os pontos <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img79.svg" alt="$A'$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img711.svg" alt="$B'$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img712.svg" alt="$P'$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img713.svg" alt="$V'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img714.svg" alt="$M'$" loading="lazy"></span> imagem pela <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> dos pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img4.svg" alt="$P$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img310.svg" alt="$V$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img101.svg" alt="$M$" loading="lazy"></span>
            respectivamente; e as retas <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img715.svg" alt="$r'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img716.svg" alt="$s'$" loading="lazy"></span> imagem pela <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span> das retas <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span> respectivamente, conforme figura
            <a href="#figtrig2">2.4</a>.
            </p>
            <div class="CENTER"><a id="figtrig2"></a><a id="1946"></a>
            <table id="2I4" title="2I4">
            <caption class="BOTTOM"><strong>Figura 2.4:</strong>
             Propriedades adicionais na hipérbole. </caption>
            <tbody><tr><td>
            <div class="CENTER">
            <img src="img/fighip02.png" alt="Image fighip02" loading="lazy"> </div></td></tr>
            </tbody></table>
            </div>
            
            <p class=" unidade" id="2P27" title="2P27">
            Nestes termos, como os pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img4.svg" alt="$P$" loading="lazy"></span> estão sobre uma mesma reta, pela propriedade <!-- MATH
             $\mathbf{(b)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img646.svg" alt="$\mathbf{(b)}$" loading="lazy"></span> da Proposição
            <a href="#prophip">2.1</a> os pontos <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img79.svg" alt="$A'$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img711.svg" alt="$B'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img712.svg" alt="$P'$" loading="lazy"></span> estão também sobre uma mesma reta. Pela mesma razão, os pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img717.svg" alt="$O$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img714.svg" alt="$M'$" loading="lazy"></span>,
            <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img713.svg" alt="$V'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img712.svg" alt="$P'$" loading="lazy"></span> também estão alinhados, isto é, sobre uma reta. Os pontos <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img79.svg" alt="$A'$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img713.svg" alt="$V'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img711.svg" alt="$B'$" loading="lazy"></span> ainda estão sobre a hipérbole.
            A razão entre as medidas dos segmentos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img68.svg" alt="$AP$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img718.svg" alt="$PB$" loading="lazy"></span> é igual 1 e, portanto, pelo item <!-- MATH
             $\mathbf{(c)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img647.svg" alt="$\mathbf{(c)}$" loading="lazy"></span> da Proposição
            <a href="#prophip">2.1</a>, a razão entre as medidas dos segmentos <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img719.svg" alt="$A'P'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img720.svg" alt="$P'B'$" loading="lazy"></span> é também 1, isto é, o ponto <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img712.svg" alt="$P'$" loading="lazy"></span> é ainda o ponto
            médio do segmento <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img721.svg" alt="$A'B'$" loading="lazy"></span>. Pelas propriedades <!-- MATH
             $\mathbf{(d)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img648.svg" alt="$\mathbf{(d)}$" loading="lazy"></span> e <!-- MATH
             $\mathbf{(e)}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img649.svg" alt="$\mathbf{(e)}$" loading="lazy"></span> da mesma Proposição, as retas <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img715.svg" alt="$r'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img716.svg" alt="$s'$" loading="lazy"></span>
            são paralelas as retas imagens das assíntotas <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img706.svg" alt="$y = \pm x$" loading="lazy"></span> por <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img643.svg" alt="$T$" loading="lazy"></span>. Como as assíntotas não são alteradas, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img715.svg" alt="$r'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img716.svg" alt="$s'$" loading="lazy"></span>
            ainda são paralelas as assíntotas. Isto significa que o triângulo <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img722.svg" alt="$A'B'M'$" loading="lazy"></span> é ainda um triângulo retângulo em <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img714.svg" alt="$M'$" loading="lazy"></span>.
             </p>

:::

## 2.2 A trigonometria hiperbólica {#SECTION00620000000000000000}

::: {.raw_html}
                
                <p class=" unidade" id="2P28" title="2P28">
                A trigonometria hiperbólica é construída sobre a hipérbole trigonométrica, isto é, o braço da hipérbole de equação
                <!-- MATH
                 $x^{2} - y^{2} = 1$
                 -->
                <span class="MATH"><img style="height: 2.42ex; vertical-align: -0.50ex; " src="img/img704.svg" alt="$x^{2} - y^{2} = 1$" loading="lazy"></span>, representado na figura <a href="#fighiptrig">2.2</a>. Dado um número real <span class="MATH"><img style="height: 1.94ex; vertical-align: -0.38ex; " src="img/img46.svg" alt="$u \geq 0$" loading="lazy"></span>, entenderemos por ângulo
                hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, ou ângulo hiperbólico <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img723.svg" alt="$VOA$" loading="lazy"></span>, o arco <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img36.svg" alt="$VA$" loading="lazy"></span> da hipérbole no primeiro quadrante, de forma que a área do
                setor <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img60.svg" alt="$OVA$" loading="lazy"></span> seja igual a <!-- MATH
                 $\frac{u}{2}$
                 -->
                <span class="MATH"><img style="height: 2.45ex; vertical-align: -0.83ex; " src="img/img61.svg" alt="$\frac{u}{2}$" loading="lazy"></span> (Ver figura <a href="#figanghippos">2.5</a>). No caso em que <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img52.svg" alt="$u < 0$" loading="lazy"></span> o ângulo hiperbólico
                <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> é o arco <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img36.svg" alt="$VA$" loading="lazy"></span> da hipérbole, no quarto quadrante, de forma que a área do setor <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img60.svg" alt="$OVA$" loading="lazy"></span> seja igual a <!-- MATH
                 $\frac{-u}{2}$
                 -->
                <span class="MATH"><img style="height: 2.64ex; vertical-align: -0.83ex; " src="img/img724.svg" alt="$\frac{-u}{2}$" loading="lazy"></span>.
                <a name="1964"></a></p>
                
                <a id="figanghippos"></a><a id="1967"></a>
                    <div id="2I5" title="2I5" class="interativo unidade">
                        <div class="controles_interatividade">
                        
                        <a href="interativo/fig2-5.html" target="_blank" class="btn_abrirInterativo">Ver maior</a><span class="barra"> | </span><span class="referencia" onclick="alert('A referência é: 2I5.')">Referência</span>
                        
                        </div>
                        
                        <iframe id="fig2-5" class="graficos" loading="lazy" src="interativo/fig2-5.html"></iframe>
                        </div>
                
                <p class=" unidade" id="2P29" title="2P29">
                Note que esta definição, em termos de área, é escolhida pois o deslizamento hiperbólico não altera área de figuras
                no plano (Proposição <a href="#nochangearea">2.2</a>) e desta forma um ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> não será alterado quando aplicarmos o
                deslizamento hiperbólico. Na figura <a href="#figanghipdes">2.6</a>, o ângulo hiperbólico <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img725.svg" alt="$OAV$" loading="lazy"></span> é igual ao ângulo hiperbólico
                <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img726.svg" alt="$OA'V'$" loading="lazy"></span> se <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img79.svg" alt="$A'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img713.svg" alt="$V'$" loading="lazy"></span> são imagens respectivas dos pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img310.svg" alt="$V$" loading="lazy"></span> por deslizamento hiperbólico.
                </p>
                <div class="CENTER"><a id="figanghipdes"></a><a id="1974"></a>
                <table id="2I6" title="2I6">
                <caption class="BOTTOM"><strong>Figura 2.6:</strong>
                Ângulo hiperbólico deslizado.</caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipdes.png" alt="Image anghipdes" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P30" title="2P30">
                Note ainda que podemos considerar ângulos hiperbólicos de qualquer magnitude, já que a área do setor entre as
                assíntotas <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img706.svg" alt="$y = \pm x$" loading="lazy"></span> e a hipérbole, é infinita. Vamos agora definir seno e cosseno hiperbólico de um ângulo
                (hiperbólico) <!-- MATH
                 $u \in \mathbb{R}$
                 -->
                <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>.
                </p>
                <p class=" unidade" id="2P31" title="2P31">
                Nesses termos, dado um ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, determinado pelo arco de hipérbole <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img36.svg" alt="$VA$" loading="lazy"></span>, consideramos o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img4.svg" alt="$P$" loading="lazy"></span>,
                projeção do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> sobre o eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span> e o ponto <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img6.svg" alt="$Q$" loading="lazy"></span> projeção do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> sobre o eixo <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img71.svg" alt="$y$" loading="lazy"></span>. O cosseno hiperbólico de
                <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span><a name="1977"></a> é definido como sendo a abscissa do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>, isto é, o comprimento do segmento
                orientado <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img74.svg" alt="$OP$" loading="lazy"></span> (ou <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img75.svg" alt="$QA$" loading="lazy"></span>), com relação ao eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>. Note que este segmento orientado nunca terá sentido contrário ao
                eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span> e, portanto, a medida de cosseno hiperbólico de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> será sempre positiva (maior ou igual a 1 para ser mais
                preciso). O seno hiperbólico de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span><a name="1978"></a> é definido como sendo a ordenada do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>, isto é, o
                comprimento do segmento orientado <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img73.svg" alt="$PA$" loading="lazy"></span> (ou <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img72.svg" alt="$OQ$" loading="lazy"></span>), com relação ao eixo <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img71.svg" alt="$y$" loading="lazy"></span>, isto é, se o segmento orientado <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img73.svg" alt="$PA$" loading="lazy"></span> tem
                sentido contrário ao eixo <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img71.svg" alt="$y$" loading="lazy"></span>, então entendemos que a medida do segmento é negativa. Isto ocorrerá apenas para valores
                negativos de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>.
                </p>
               <a id="figsencoship"></a><a id="1981"></a>
               <div id="2I7" title="2I7" class="interativo unidade">
                <div class="controles_interatividade">
                
                <a href="interativo/fig2-7.html" target="_blank" class="btn_abrirInterativo">Ver maior</a><span class="barra"> | </span><span class="referencia" onclick="alert('A referência é: 2I7.')">Referência</span>
                
                </div>
                
                <iframe id="fig2-7" class="graficos" loading="lazy" src="interativo/fig2-7.html"></iframe>
                </div>
                
                
                <p class=" unidade" id="2P32" title="2P32">
                Representamos isto escrevendo
                </p><!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}u = PA \qquad \text{e} \qquad \cosh u = OP.
                \end{displaymath}
                 -->
               
                <div class="mathdisplay unidade" id="2P33" title="2P33">
                <img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img727.svg" alt="$\displaystyle {\mathrm{senh}}u = PA$" loading="lazy">&nbsp; &nbsp;e<img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img728.svg" alt="$\displaystyle \qquad \cosh u = OP. $" loading="lazy">
                </div>
                
                <p class=" unidade" id="2P34" title="2P34">
                As demais funções trigonométricas hiperbólicas, tangente, cotangente, secante e cossecante, são definidas como na
                trigonometria circular, isto é, respectivamente <a name="1985"></a><a name="1986"></a><a name="1987"></a><a name="1988"></a>
                </p>
                <div class="mathdisplay unidade" id="2P35" title="2P35"><table class="equation">
                <tbody><tr>
                <td style="text-align:right;"><span class="MATH"><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"></span></td>
                <td style="text-align:left;"><span class="MATH">&nbsp;<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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td style="text-align:right;"><span class="MATH"><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"></span></td>
                <td style="text-align:left;"><span class="MATH">e<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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                </tbody></table></div>
                
                
                <p class=" unidade" id="2P36" title="2P36">
                Se considerarmos dois ângulos hiperbólicos de medidas <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img78.svg" alt="$-u$" loading="lazy"></span>, representados respectivamente pelos arcos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img36.svg" alt="$VA$" loading="lazy"></span> e
                <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img733.svg" alt="$VC$" loading="lazy"></span>, vemos (na figura <a href="#figangcont">2.8</a>) que os valores de seno hiperbólico são diferentes apenas por um sinal, pois os
                segmentos orientados <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img72.svg" alt="$OQ$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.21ex; vertical-align: -0.50ex; " src="img/img734.svg" alt="$OQ'$" loading="lazy"></span> tem sentidos opostos e que os valores de cosseno hiperbólico são ambos iguais ao
                segmento orientado <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img74.svg" alt="$OP$" loading="lazy"></span>.
                </p>
                <div class="CENTER"><a id="figangcont"></a><a id="2004"></a>
                <table id="2I8" title="2I8">
                <caption class="BOTTOM"><strong>Figura 2.8:</strong>
                Ângulos hiperbólicos de sinais contrários.</caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/sencoshpn.png" alt="Image sencoshpn" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P37" title="2P37">
                Isto significa que,
                </p>
                <div class="mathdisplay unidade" id="2P38" title="2P38"><a id="sencoshparimpar"></a><!-- MATH
                 \begin{equation}
                {\mathrm{senh}}u = -{\mathrm{senh}}(-u), \qquad \text{e} \qquad \cosh u = \cosh(-u).
                \end{equation}
                 -->
                <table>
                <tbody><tr>
                <td style="text-align:center;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img735.svg" alt="$\displaystyle {\mathrm{senh}}u = -{\mathrm{senh}}(-u),$" loading="lazy">&nbsp; &nbsp;e<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img736.svg" alt="$\displaystyle \qquad \cosh u = \cosh(-u).$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right; vertical-align: middle;">
                (<span class="arabic">2</span>.<span class="arabic">2</span>)</td></tr>
                </tbody></table></div>
                <p style="text-indent:0px !important;" class=" unidade" id="2P39" title="2P39">
                Em outras palavras, o seno hiperbólico é uma função ímpar e o cosseno hiperbólico é uma função par.
                </p>
                <p class=" unidade" id="2P40" title="2P40">
                Nosso próximo passo é deduzir as principais fórmulas da trigonometria hiperbólica. Serão cinco fórmulas, contando com
                as duas identidades em (<a href="#sencoshparimpar">2.2</a>). Faltam a relação fundamental e as fórmulas de soma de arcos para o seno
                e o cosseno hiperbólicos. Demais fórmulas trigonométricas que se deseje podem ser deduzidas a partir destas cinco.
                </p>
                <p class=" unidade" id="2P41" title="2P41">
                Da figura <a href="#figsencoship">2.7</a>, podemos ver claramente que as coordenadas cartesianas do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> são <!-- MATH
                 $A = (\cosh u,
                {\mathrm{senh}}u)$
                 -->
                <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img737.svg" alt="$A = (\cosh u,
                {\mathrm{senh}}u)$" loading="lazy"></span>. Também o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> está sobre a hipérbole, e então suas coordenadas devem obrigatoriamente satisfazer a
                equação da hipérbole <!-- MATH
                 $x^{2} - y^{2} = 1$
                 -->
                <span class="MATH"><img style="height: 2.42ex; vertical-align: -0.50ex; " src="img/img704.svg" alt="$x^{2} - y^{2} = 1$" loading="lazy"></span> e, assim,
                </p>
                <div class="mathdisplay unidade" id="2P42" title="2P42"><a id="idfundhip"></a><!-- MATH
                 \begin{equation}
                \cosh^{2} u - {\mathrm{senh}}^{2} u = (\cosh u)^{2} - ({\mathrm{senh}}u)^{2} = 1,
                \end{equation}
                 -->
                <table >
                <tbody><tr>
                <td style="text-align:center;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right; vertical-align: middle;">
                (<span class="arabic">2</span>.<span class="arabic">3</span>)</td></tr>
                </tbody></table></div>
                <p style="text-indent:0px !important;" class=" unidade" id="2P43" title="2P43">
                que é a relação fundamental da trigonometria hiperbólica.
                </p>
                <p class=" unidade" id="2P44" title="2P44">
                Vamos agora mostrar a validade das fórmulas trigonométricas da soma de arcos do cosseno hiperbólico e do seno
                hiperbólico. Consideremos dois ângulos hiperbólicos <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img2.svg" alt="$v$" loading="lazy"></span> determinados pelos arcos hiperbólicos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img739.svg" alt="$VZ$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img54.svg" alt="$VB$" loading="lazy"></span>
                respectivamente, conforme a figura abaixo.
                </p>
                <div class="CENTER"><a id="figanghipuv"></a><a id="2024"></a>
                <table id="2I9" title="2I9">
                <caption class="BOTTOM"><strong>Figura 2.9:</strong>
                Ângulos hiperbólicos <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img2.svg" alt="$v$" loading="lazy"></span>.</caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipuv.png" alt="Image anghipuv" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P45" title="2P45">
                Tomando o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>, sobre a hipérbole, simétrico do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img740.svg" alt="$Z$" loading="lazy"></span> pelo eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>, temos o segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img741.svg" alt="$AZ$" loading="lazy"></span> conjugado ao
                segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span>, isto é, o prolongamento do segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span> corta o segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img741.svg" alt="$AZ$" loading="lazy"></span> em seu ponto médio. Chamemos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img742.svg" alt="$W$" loading="lazy"></span>, este
                ponto médio. <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img742.svg" alt="$W$" loading="lazy"></span> também é a projeção de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img740.svg" alt="$Z$" loading="lazy"></span> sobre o eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>. Considerando as retas <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span> paralelas as assíntotas
                <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img706.svg" alt="$y = \pm x$" loading="lazy"></span>, que passam pelos pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img740.svg" alt="$Z$" loading="lazy"></span> respectivamente, temos que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span> se encontram sobre o eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span> no
                ponto que denotaremos por <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img101.svg" alt="$M$" loading="lazy"></span>.
                </p>
                <div class="CENTER"><a id="2029"></a>
                <table id="2I10" title="2I10">
                <caption class="BOTTOM"><strong>Figura 2.10:</strong>
                 Segmento conjugado ao ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>.</caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipuv2.png" alt="Image anghipuv2" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P46" title="2P46">
                Decorre disto que
                </p><!-- MATH
                 \begin{displaymath}
                \cosh u = OW \qquad \text{e} \qquad {\mathrm{senh}}u = WZ.
                \end{displaymath}
                 -->
                
                <div class="mathdisplay unidade" id="2P47" title="2P47">
                <img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img743.svg" alt="$\displaystyle \cosh u = OW$" loading="lazy">&nbsp; &nbsp;e<img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img744.svg" alt="$\displaystyle \qquad {\mathrm{senh}}u = WZ. $" loading="lazy">
                </div>
                
                <p class=" unidade" id="2P48" title="2P48">
                Vamos agora aplicar um deslizamento hiperbólico que desliza o arco <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img739.svg" alt="$VZ$" loading="lazy"></span> de forma que a imagem <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img713.svg" alt="$V'$" loading="lazy"></span> de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img310.svg" alt="$V$" loading="lazy"></span> coincide com
                o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span>. A imagem de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img740.svg" alt="$Z$" loading="lazy"></span> então será denotada por <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img133.svg" alt="$C$" loading="lazy"></span>, isto é, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img745.svg" alt="$Z' = C$" loading="lazy"></span>. Lembrando ainda que o ângulo hiperbólico
                <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img746.svg" alt="$BOC$" loading="lazy"></span> continua sendo o ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, em virtude da invariância de áreas por deslizamento hiperbólico. O ponto
                <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img133.svg" alt="$C$" loading="lazy"></span> por sua vez determina o arco de hipérbole <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img733.svg" alt="$VC$" loading="lazy"></span> associado ao ângulo hiperbólico <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img135.svg" alt="$(u+v)$" loading="lazy"></span>. Também, sejam <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img79.svg" alt="$A'$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img747.svg" alt="$W'$" loading="lazy"></span> e
                <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img714.svg" alt="$M'$" loading="lazy"></span> as respectivas imagens dos pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img742.svg" alt="$W$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img101.svg" alt="$M$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img715.svg" alt="$r'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img716.svg" alt="$s'$" loading="lazy"></span> as respectivas imagens das retas <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img40.svg" alt="$r$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img118.svg" alt="$s$" loading="lazy"></span>.
                </p>
                <div class="CENTER"><a id="figanghipu"></a><a id="2034"></a>
                <table id="2I11" title="2I11">
                <caption class="BOTTOM"><strong>Figura 2.11:</strong>
                 Ângulo hiperbólico <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img3.svg" alt="$u+v$" loading="lazy"></span>. </caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipuv3.png" alt="Image anghipuv3" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P49" title="2P49">
                Pelos pontos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img53.svg" alt="$B$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img133.svg" alt="$C$" loading="lazy"></span> traçamos, as perpendiculares ao eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span>, <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img139.svg" alt="$BP$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img748.svg" alt="$CR$" loading="lazy"></span>. Lembremos que a corda <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img749.svg" alt="$CA'$" loading="lazy"></span>, é ainda
                conjugada a <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img750.svg" alt="$OB$" loading="lazy"></span>, ou seja, o prolongamento de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img750.svg" alt="$OB$" loading="lazy"></span> encontra o ponto médio do segmento <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img749.svg" alt="$CA'$" loading="lazy"></span> e com <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img747.svg" alt="$W'$" loading="lazy"></span> sendo este
                ponto médio.
                </p>
                <div class="CENTER"><a id="2039"></a>
                <table id="2I12" title="2I12">
                <caption class="BOTTOM"><strong>Figura 2.12:</strong>
                 Projeções <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img4.svg" alt="$P$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img5.svg" alt="$R$" loading="lazy"></span>. </caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipuv4.png" alt="Image anghipuv4" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P50" title="2P50">
                Nestes termos temos as relações,
                </p><!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}v = PB \qquad \text{e} \qquad \cosh v = OP,
                \end{displaymath}
                 -->
                
                <div class="mathdisplay unidade" id="2P51" title="2P51">
                <img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img751.svg" alt="$\displaystyle {\mathrm{senh}}v = PB$" loading="lazy">&nbsp; &nbsp;e<img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img752.svg" alt="$\displaystyle \qquad \cosh v = OP, $" loading="lazy">
                </div><p class=" unidade" id="2P52" style="text-indent: 0 !important;" title="2P52">
                e
            </p><!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}(u+v) = RC \qquad \text{e} \qquad \cosh(u+v) = OR.
                \end{displaymath}
                 -->
                
                <div class="mathdisplay unidade" id="2P53" title="2P53">
                <img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img753.svg" alt="$\displaystyle {\mathrm{senh}}(u+v) = RC$" loading="lazy">&nbsp; &nbsp;e<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img754.svg" alt="$\displaystyle \qquad \cosh(u+v) = OR. $" loading="lazy">
                </div>
                
                <p class=" unidade" id="2P54" title="2P54">
                Vamos agora mostrar que também valem,
                </p>
                <div class="mathdisplay unidade" id="2P55" title="2P55"><a id="idsencosh"></a><!-- MATH
                 \begin{equation}
                {\mathrm{senh}}u = \frac{W'C}{OB} \qquad \text{e} \qquad \cosh u = \frac{OW'}{OB}.
                \end{equation}
                 -->
                <table>
                <tbody><tr>
                <td style="text-align:center;"><span class="MATH"><img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img755.svg" alt="$\displaystyle {\mathrm{senh}}u = \frac{W'C}{OB}$" loading="lazy">&nbsp; &nbsp;e<img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img756.svg" alt="$\displaystyle \qquad \cosh u = \frac{OW'}{OB}.$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right; vertical-align: middle;">
                (<span class="arabic">2</span>.<span class="arabic">4</span>)</td></tr>
                </tbody></table></div>
                
                
                <p class=" unidade" id="2P56" title="2P56">
                Os segmentos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img757.svg" alt="$OW$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span> estão sobre a mesma reta e então do item <!-- MATH
                 $\mathbf{(c)}$
                 -->
                <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img647.svg" alt="$\mathbf{(c)}$" loading="lazy"></span> da proposição <a href="#prophip">2.1</a> temos que
                a razão <!-- MATH
                 $\frac{OW}{OV}$
                 -->
                <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img758.svg" alt="$\frac{OW}{OV}$" loading="lazy"></span> é preservada pelo deslizamento hiperbólico, ou seja, <!-- MATH
                 $\frac{OW}{OV} = \frac{OW'}{OB}$
                 -->
                <span class="MATH"><img style="height: 2.93ex; vertical-align: -0.83ex; " src="img/img759.svg" alt="$\frac{OW}{OV} = \frac{OW'}{OB}$" loading="lazy"></span>. Levando
                em conta que <span class="MATH"><img style="height: 1.66ex; vertical-align: -0.11ex; " src="img/img760.svg" alt="$OV = 1$" loading="lazy"></span>, temos imediatamente que
                </p><!-- MATH
                 \begin{displaymath}
                \cosh u = OW = \frac{OW}{OV} = \frac{OW'}{OB}.
                \end{displaymath}
                 -->
                
                <div class="mathdisplay unidade" id="2P57" title="2P57">
                <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">
                </div>
                
                <p class=" unidade" id="2P58" title="2P58">
                Agora, os triângulos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img762.svg" alt="$AMZ$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img763.svg" alt="$A'M'C$" loading="lazy"></span> são triângulos retângulos e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img742.svg" alt="$W$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img747.svg" alt="$W'$" loading="lazy"></span> são pontos médios das respectivas
                hipotenusas. O ponto médio da hipotenusa é equidistante aos vértices de um triângulo retângulo, isto é, <!-- MATH
                 $WA = WM = WZ$
                 -->
                <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img764.svg" alt="$WA = WM = WZ$" loading="lazy"></span>
                e <!-- MATH
                 $W'A' = W'M' = W'C$
                 -->
                <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img765.svg" alt="$W'A' = W'M' = W'C$" loading="lazy"></span>. Também, como os segmentos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img766.svg" alt="$MW$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span> estão sobre uma mesma reta, a razão <!-- MATH
                 $\frac{MW}{OV}$
                 -->
                <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img767.svg" alt="$\frac{MW}{OV}$" loading="lazy"></span> é
                preservada pelo deslizamento hiperbólico, isto é, <!-- MATH
                 $\frac{MW}{OV} = \frac{M'W'}{OV'}$
                 -->
                <span class="MATH"><img style="height: 2.93ex; vertical-align: -0.83ex; " src="img/img768.svg" alt="$\frac{MW}{OV} = \frac{M'W'}{OV'}$" loading="lazy"></span>. Segue que
                <!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}u = WZ = \frac{WZ}{OV} = \frac{MW}{OV} = \frac{M'W'}{OB} = \frac{W'C}{OB},
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P59" title="2P59">
                <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">
                </div>
                e isto garante as igualdades (<a href="#idsencosh">2.4</a>).
                
                <p class=" unidade" id="2P60" title="2P60">
                Seja <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img6.svg" alt="$Q$" loading="lazy"></span> a projeção de <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img747.svg" alt="$W'$" loading="lazy"></span> sobre o eixo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img69.svg" alt="$x$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img7.svg" alt="$S$" loading="lazy"></span> a projeção de <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img747.svg" alt="$W'$" loading="lazy"></span> sobre o segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img770.svg" alt="$RC$" loading="lazy"></span>, conforme a figura
                <a href="#figanghipuv2">2.13</a>.
                </p>
                <div class="CENTER"><a id="figanghipuv2"></a><a id="2081"></a>
                <table id="2I13" title="2I13">
                <caption class="BOTTOM"><strong>Figura 2.13:</strong>
                 Projeções <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img6.svg" alt="$Q$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img7.svg" alt="$S$" loading="lazy"></span>.</caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipuv5.png" alt="Image anghipuv5" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P61" title="2P61">
                Notemos que os triângulos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img771.svg" alt="$OPB$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.21ex; vertical-align: -0.50ex; " src="img/img772.svg" alt="$OQW'$" loading="lazy"></span> são semelhantes. Vamos verificar que também são semelhantes os triângulos
                <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img771.svg" alt="$OPB$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img773.svg" alt="$CSW'$" loading="lazy"></span>. Para isto mostraremos que o ângulo <span class="MATH"><img style="height: 2.26ex; vertical-align: -0.10ex; " src="img/img774.svg" alt="$B\hat{O}P$" loading="lazy"></span> é igual ao ângulo <!-- MATH
                 $S\hat{C}W'$
                 -->
                <span class="MATH"><img style="height: 2.26ex; vertical-align: -0.10ex; " src="img/img775.svg" alt="$S\hat{C}W'$" loading="lazy"></span>. A reta <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img716.svg" alt="$s'$" loading="lazy"></span>, paralela a
                bissetriz <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img708.svg" alt="$y=x$" loading="lazy"></span>, passa por <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img133.svg" alt="$C$" loading="lazy"></span>, intercepta <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img750.svg" alt="$OB$" loading="lazy"></span> em <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img714.svg" alt="$M'$" loading="lazy"></span> e intercepta <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img132.svg" alt="$OV$" loading="lazy"></span> em um ponto que chamaremos de <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img8.svg" alt="$N$" loading="lazy"></span> (Figura
                <a href="#figanghipuv3">2.14</a>).
                </p>
                <div class="CENTER"><a id="figanghipuv3"></a><a id="2089"></a>
                <table id="2I14" title="2I14">
                <caption class="BOTTOM"><strong>Figura 2.14:</strong>
                Projeção <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img8.svg" alt="$N$" loading="lazy"></span>.</caption>
                <tbody><tr><td>
                <div class="CENTER">
                <img src="img/anghipuv6.png" alt="Image anghipuv6" loading="lazy"> </div></td></tr>
                </tbody></table>
                </div>
                
                <p class=" unidade" id="2P62" title="2P62">
                Assim, <!-- MATH
                 $R\hat{C}N = R\hat{N}C = 45^{\circ}$
                 -->
                <span class="MATH"><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"></span>. Mais ainda, como o triângulo <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img777.svg" alt="$CM'A'$" loading="lazy"></span> é retângulo em <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img714.svg" alt="$M'$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img747.svg" alt="$W'$" loading="lazy"></span> é o ponto
                médio da hipotenusa <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img749.svg" alt="$CA'$" loading="lazy"></span>, segue que o triângulo <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img778.svg" alt="$M'W'C$" loading="lazy"></span> é isósceles e pontanto os ângulos <!-- MATH
                 $W'\hat{M'}C$
                 -->
                <span class="MATH"><img style="height: 2.33ex; vertical-align: -0.10ex; " src="img/img779.svg" alt="$W'\hat{M'}C$" loading="lazy"></span> e
                <!-- MATH
                 $W'\hat{C}M'$
                 -->
                <span class="MATH"><img style="height: 2.26ex; vertical-align: -0.10ex; " src="img/img780.svg" alt="$W'\hat{C}M'$" loading="lazy"></span> possuem a mesma medida, isto é, <!-- MATH
                 $W'\hat{M'}C = W'\hat{C}M'$
                 -->
                <span class="MATH"><img style="height: 2.33ex; vertical-align: -0.10ex; " src="img/img781.svg" alt="$W'\hat{M'}C = W'\hat{C}M'$" loading="lazy"></span>. Mas <!-- MATH
                 \begin{displaymath}
                M'\hat{N}P + M'\hat{N}O = 180^{\circ}
                = M'\hat{N}O + M'\hat{O}N + N\hat{M'}O, 
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P63" title="2P63">
                <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}
                = M'\hat{N}O + M'\hat{O}N + N\hat{M'}O, $" loading="lazy">
                </div><p class=" unidade" id="2P64" style="text-indent: 0;" title="2P64">
                e, portanto, <!-- MATH
                 $M'\hat{N}P = N\hat{M'}O + M'\hat{O}N = N\hat{M'}O + M'\hat{O}P$
                 -->
                <span class="MATH"><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"></span>.
                Segue disto que
                <!-- MATH
                 \begin{displaymath}
                B\hat{O}P = M'\hat{O}P = M'\hat{N}P - N\hat{M'}O = C\hat{N}R - C\hat{M'}W'.
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P65" title="2P65">
                <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">
                </div>
                
                <p class=" unidade" id="2P66" style="text-indent: 0 !important;" title="2P66">
                Agora, <!-- MATH
                 $S\hat{C}W' = R\hat{C}N - M'\hat{C}W'$
                 -->
                <span class="MATH"><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"></span> e, portanto,
                <!-- MATH
                 \begin{displaymath}
                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,
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P67" title="2P67">
                <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">
                </div><p class=" unidade" id="2P68" style="text-indent: 0 !important;" title="2P68">
                e como <!-- MATH
                 $C\hat{N}R = R\hat{C}N$
                 -->
                <span class="MATH"><img style="height: 2.26ex; vertical-align: -0.10ex; " src="img/img787.svg" alt="$C\hat{N}R = R\hat{C}N$" loading="lazy"></span> então temos,
                <!-- MATH
                 \begin{displaymath}
                B\hat{O}P = S\hat{C}W',
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P69" title="2P69">
                <img style="height: 2.26ex; vertical-align: -0.10ex; " src="img/img788.svg" alt="$\displaystyle B\hat{O}P = S\hat{C}W', $" loading="lazy">
                </div><p class=" unidade" id="2P70" style="text-indent: 0 !important;" title="2P70">
                como desejado. Isto mostra que os triângulos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img789.svg" alt="$BOP$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.81ex; vertical-align: -0.10ex; " src="img/img773.svg" alt="$CSW'$" loading="lazy"></span> são semelhantes. Desta semelhança, segue que
                <!-- MATH
                 \begin{displaymath}
                \frac{SC}{CW'} = \frac{OP}{OB} \qquad \text{e} \qquad \frac{W'S}{W'C} = \frac{PB}{OB}
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P71" title="2P71">
                <img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img790.svg" alt="$\displaystyle \frac{SC}{CW'} = \frac{OP}{OB}$" loading="lazy">&nbsp; &nbsp;e<img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img791.svg" alt="$\displaystyle \qquad \frac{W'S}{W'C} = \frac{PB}{OB} $" loading="lazy">
                </div><p class=" unidade" id="2P72" style="text-indent: 0 !important;" title="2P72">
                e destas igualdades,
                <!-- MATH
                 \begin{displaymath}
                SC = \frac{CW'}{OB} OP \qquad \text{e} \qquad W'S = \frac{W'C}{OB} PB.
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P73" title="2P73">
                <img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img792.svg" alt="$\displaystyle SC = \frac{CW'}{OB} OP$" loading="lazy">&nbsp; &nbsp;e<img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img793.svg" alt="$\displaystyle \qquad W'S = \frac{W'C}{OB} PB. $" loading="lazy">
                </div>
                
                <p class=" unidade" id="2P74" title="2P74">
                Também são semelhantes os triângulos <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img789.svg" alt="$BOP$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.21ex; vertical-align: -0.50ex; " src="img/img794.svg" alt="$W'OQ$" loading="lazy"></span> e desta semelhança, temos
                <!-- MATH
                 \begin{displaymath}
                \frac{QW'}{OW'} = \frac{PB}{OB} \qquad \text{e} \qquad \frac{OQ}{OW'} = \frac{OP}{OB}
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P75" title="2P75">
                <img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img795.svg" alt="$\displaystyle \frac{QW'}{OW'} = \frac{PB}{OB}$" loading="lazy">&nbsp; &nbsp;e<img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img796.svg" alt="$\displaystyle \qquad \frac{OQ}{OW'} = \frac{OP}{OB} $" loading="lazy">
                </div><p class=" unidade" id="2P76" style="text-indent: 0 !important;" title="2P76">
                e destas igualdades,
                <!-- MATH
                 \begin{displaymath}
                QW' = \frac{OW'}{OB} PB \qquad \text{e} \qquad OQ = \frac{OW'}{OB} OP.
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P77" title="2P77">
                <img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img797.svg" alt="$\displaystyle QW' = \frac{OW'}{OB} PB$" loading="lazy">&nbsp; &nbsp;e<img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img798.svg" alt="$\displaystyle \qquad OQ = \frac{OW'}{OB} OP. $" loading="lazy">
                </div>
                
                <p class=" unidade" id="2P78" title="2P78">
                Finalmente, lembrando que
                <!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}v = PB, \qquad \cosh v = OP,
                \end{displaymath}
                 -->
                </p>
                <div class="mathdisplay unidade" id="2P79" title="2P79">
                <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">
                </div>
                <!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}u = \frac{W'C}{OB}, \qquad \cosh u = \frac{OW'}{OB},
                \end{displaymath}
                 -->
                
                <div class="mathdisplay unidade" id="2P80" title="2P80">
                <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">
                </div>
                <!-- MATH
                 \begin{displaymath}
                {\mathrm{senh}}(u+v) = RC \qquad \text{e} \qquad \cosh(u+v) = OR,
                \end{displaymath}
                 -->
                
                <div class="mathdisplay unidade" id="2P81" title="2P81">
                <img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img753.svg" alt="$\displaystyle {\mathrm{senh}}(u+v) = RC$" loading="lazy">&nbsp; &nbsp;e<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img801.svg" alt="$\displaystyle \qquad \cosh(u+v) = OR, $" loading="lazy">
                </div><p class=" unidade" id="2P82" style="text-indent: 0 !important;" title="2P82">
                temos,
                </p>
                <div class="mathdisplay unidade" id="2P83" title="2P83"><table>
                <tbody><tr>
                <td style="text-align:right;width: 39%"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img802.svg" alt="$\displaystyle {\mathrm{senh}}(u+v)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.31ex; vertical-align: -0.50ex; " src="img/img803.svg" alt="$\displaystyle = RC = QW' + SC$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img804.svg" alt="$\displaystyle = \frac{CW'}{OB} OP + \frac{OW'}{OB} PB$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 1.83ex; vertical-align: -0.27ex; " src="img/img805.svg" alt="$\displaystyle = {\mathrm{senh}}u \cosh v + \cosh u {\mathrm{senh}}v,$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                <a id="addarcsenh">(<span class="arabic">2</span>.<span class="arabic">5</span>)</a></td></tr>
                </tbody></table></div>
                <p class=" unidade" id="2P84" style="text-indent: 0 !important;" title="2P84">
                e também,
                </p>
                <div class="mathdisplay unidade" id="2P85" title="2P85"><table>
                <tbody><tr>
                <td style="text-align:right; width: 39%;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img806.svg" alt="$\displaystyle \cosh(u+v)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.31ex; vertical-align: -0.50ex; " src="img/img807.svg" alt="$\displaystyle = OR = OQ + W'S$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 4.68ex; vertical-align: -1.55ex; " src="img/img808.svg" alt="$\displaystyle = \frac{OW'}{OB} OP + \frac{W'C}{OB} PB$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 1.83ex; vertical-align: -0.27ex; " src="img/img809.svg" alt="$\displaystyle = \cosh u \cosh v + {\mathrm{senh}}u {\mathrm{senh}}v.$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                <a id="addarccosh">(<span class="arabic">2</span>.<span class="arabic">6</span>)</a></td></tr>
                </tbody></table></div>
                
                
                <p class=" unidade" id="2P86" title="2P86">
                As fórmulas (<a href="#addarcsenh">2.5</a>) e (<a href="#addarccosh">2.6</a>), juntamente com a relação fundamental (<a href="#idfundhip">2.3</a>) e as duas
                fórmulas em (<a href="#sencoshparimpar">2.2</a>), constituem as 5 fórmulas básicas da trigonometria hiperbólica. Com elas podemos
                deduzir outras fórmulas, como por exemplo, as fórmulas de duplicação de arcos,
                </p>
                <div class="mathdisplay unidade" id="2P87" title="2P87"><table class="equation">
                <tbody><tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img810.svg" alt="$\displaystyle \cosh(2u)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img811.svg" alt="$\displaystyle = \cosh(u+u)$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.27ex; " src="img/img812.svg" alt="$\displaystyle = \cosh^{2} u + {\mathrm{senh}}^{2} u$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
               
                <tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img813.svg" alt="$\displaystyle {\mathrm{senh}}(2u)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img814.svg" alt="$\displaystyle = {\mathrm{senh}}(u+u)$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img815.svg" alt="$\displaystyle = 2{\mathrm{senh}}u \cosh u,$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                </tbody></table></div>
                <p class=" unidade" id="2P88" style="text-indent: 0 !important;" title="2P88">
                e as fórmulas de diferença de arcos,
                </p>
                <div class="mathdisplay unidade" id="2P89" title="2P89"><table class="equation">
                <tbody><tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img816.svg" alt="$\displaystyle \cosh(u - v)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img819.svg" alt="$\displaystyle {\mathrm{senh}}(u - v)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                </tbody></table></div>
                
                
                <p class=" unidade" id="2P90" title="2P90">
                Vamos agora obter duas outras fórmulas trigonométricas hiperbólicas que serão úteis mais adiante. São fórmulas fáceis
                de serem obtidas, similares às fórmulas obtidas na proposição <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#propidsectg">1.1</a>. Estamos apresentando-as em virtude do
                uso futuro (na seção <a href="#secderhipinv">2.8</a>).
                </p>
                <div id="2Teo3" title="2Teo3" class=" unidade"><a id="propidsectghip"><b>Proposição  <span class="arabic">2</span>..<span class="arabic">3</span></b></a> &nbsp; 
                <i>São válidas as seguintes identidades trigonométricas hiperbólicas
                </i><table width="90%">
                <tbody><tr><td align="right" valign="top"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img184.svg" alt="$(i)$" loading="lazy"></span></td><td valign="top">&nbsp;Para todos <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img2.svg" alt="$v$" loading="lazy"></span> reais,
                
                <div class="mathdisplay unidade" id="2P91" title="2P91"><a id="idtghadd"></a><!-- MATH
                 \begin{equation}
                {\mathrm {tgh}}(u+v) = \frac{{\mathrm {tgh}}u + {\mathrm {tgh}}v}{1 + {\mathrm {tgh}}u {\mathrm {tgh}}v}.
                \end{equation}
                 -->
                <table >
                <tbody><tr>
                <td style="text-align:center;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                (<span class="arabic">2</span>.<span class="arabic">7</span>)</td></tr>
                </tbody></table></div>
                
                </td></tr>
                <tr><td align="right" valign="top"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img187.svg" alt="$(ii)$" loading="lazy"></span></td><td valign="top">&nbsp;Para todo <!-- MATH
                 $u \in \mathbb{R}$
                 -->
                <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>,
                
                <div class="mathdisplay unidade" id="2P92" title="2P92"><a id="idtghsech"></a><!-- MATH
                 \begin{equation}
                1 - {\mathrm {tgh}}^{2} u = {\mathrm{sech}}^{2} u.
                \end{equation}
                 -->
                <table >
                <tbody><tr>
                <td style="text-align:center;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                (<span class="arabic">2</span>.<span class="arabic">8</span>)</td></tr>
                </tbody></table></div>
                
                </td></tr>
                <tr><td align="right" valign="top"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img190.svg" alt="$(iii)$" loading="lazy"></span></td><td valign="top">&nbsp;Para todo <!-- MATH
                 $u \in \mathbb{R}-\{0\}$
                 -->
                <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img824.svg" alt="$u \in \mathbb{R}-\{0\}$" loading="lazy"></span>,
                
                <div class="mathdisplay unidade" id="2P93" title="2P93"><a id="idctghcsch"></a><!-- MATH
                 \begin{equation}
                {\mathrm{ctgh}}^{2} u - 1 = {\mathrm{csch}}^{2} u
                \end{equation}
                 -->
                <table >
                <tbody><tr>
                <td style="text-align:center;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                (<span class="arabic">2</span>.<span class="arabic">9</span>)</td></tr>
                </tbody></table></div>
                </td></tr></tbody></table></div>
                
                
                <div><i>Prova</i>.
                
                Para <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img184.svg" alt="$(i)$" loading="lazy"></span>, temos
                
                <div class="mathdisplay unidade" id="2P94" title="2P94"><table class="equation">
                <tbody><tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img826.svg" alt="$\displaystyle {\mathrm {tgh}}(u+v)$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 5.16ex; vertical-align: -2.07ex; " src="img/img827.svg" alt="$\displaystyle = \frac{{\mathrm{senh}}(u+v)}{\cosh(u+v)} = \frac{{\mathrm{senh}}...
                ... {\mathrm{senh}}v \cosh u}{\cosh u \cosh v + {\mathrm{senh}}u {\mathrm{senh}}v}$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 5.65ex; vertical-align: -2.52ex; " src="img/img829.svg" alt="$\displaystyle = \left( \frac{{\mathrm{senh}}u \cosh v + {\mathrm{senh}}v \cosh ...
                ...rac{1}{(1 + \frac{{\mathrm{senh}}u}{\cosh u} \frac{{\mathrm{senh}}v}{\cosh v})}$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                </tbody></table></div>
                <p class=" unidade" id="2P95" style="text-indent: 0 !important;" title="2P95">
                Os itens <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img187.svg" alt="$(ii)$" loading="lazy"></span> e <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img190.svg" alt="$(iii)$" loading="lazy"></span> ficam
                </p>
                <div class="mathdisplay unidade" id="2P96" title="2P96"><table class="equation">
                <tbody><tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.16ex; vertical-align: -0.13ex; " src="img/img833.svg" alt="$\displaystyle {\mathrm{sech}}^{2} u$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 4.79ex; vertical-align: -1.82ex; " src="img/img834.svg" alt="$\displaystyle = \frac{1}{\cosh^{2} u}$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.61ex; vertical-align: -0.58ex; " src="img/img837.svg" alt="$\displaystyle = 1- {\mathrm {tgh}}^{2} u$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                </tbody>
                </table>
                <p style="text-indent: 0 !important; text-align: left !important;"><span class="MATH">e</span></p>
                <table class="equation">
                    <tbody><tr>
                <tr>
                <td style="text-align:right;"><span class="MATH"><img style="height: 2.16ex; vertical-align: -0.13ex; " src="img/img838.svg" alt="$\displaystyle {\mathrm{csch}}^{2} u$" loading="lazy"></span></td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 4.79ex; vertical-align: -1.82ex; " src="img/img839.svg" alt="$\displaystyle = \frac{1}{{\mathrm{senh}}^{2} u}$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><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"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
                <tr>
                <td>&nbsp;</td>
                <td style="text-align:left;"><span class="MATH"><img style="height: 2.61ex; vertical-align: -0.58ex; " src="img/img842.svg" alt="$\displaystyle = {\mathrm{ctgh}}^{2} u - 1$" loading="lazy"></span></td>
                <td class="eqno" style="text-align:right">
                &nbsp;&nbsp;&nbsp;</td></tr>
            </tbody></table></div>

                <p style="text-indent: 0 !important;"><span class="MATH">e a prova está concluída.</span><span style="text-align: right;">                 <img style="height: 1.59ex; vertical-align: -0.10ex; float: right;" src="img/img193.svg" alt="$\qedsymbol$" loading="lazy">
                </span></p>
               
               
                </div>

:::

## 2.3 As funções trigonométricas hiperbólicas {#SECTION00630000000000000000}

::: {.raw_html}

              <a id="secfunchip"></a>
                    
                    <p class=" unidade" id="2P97" title="2P97">
                    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 <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> que a cada valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> associa o seno, ou o cosseno, ou a tangente, ou a cotangente, ou a secante, ou ainda a cossecante hiperbólica de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>. Vamos olhar uma a uma.
                    </p>
                    <p class=" unidade" id="2P98" title="2P98">
                    Para a função <!-- MATH
                     $w = f(u) = {\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img843.svg" alt="$w = f(u) = {\mathrm{senh}}u$" loading="lazy"></span>, notemos que para cada valor real de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, construímos o ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>
                    determinado pelo arco <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img62.svg" alt="$AV$" loading="lazy"></span>, onde a ordenada do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> é o seno hiperbólico de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>.<a name="2263"></a>
                    Não há nenhuma impossibilidade matemática para <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e, portanto, o domínio da função <!-- MATH
                     $f(u) = {\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img844.svg" alt="$f(u) = {\mathrm{senh}}u$" loading="lazy"></span> é todo o conjunto
                    dos números reais. Além disso, fazendo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> variar no conjunto dos reais, os valores resultantes para a ordenada do
                    ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> também percorrem o conjunto dos números reais. Desta forma, temos que a função
                    </p>
                     <br>
                    <div class="mathdisplay unidade" id="2P99" title="2P99">
                    <!-- MATH
                     \begin{eqnarray*}
                    f: \mathbb{R}& \to & \mathbb{R}\\
                    u & \mapsto & w = f(u) = {\mathrm{senh}}u
                    \end{eqnarray*}
                     -->
                    <table cellpadding="0" align="CENTER" width="100%">
                    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img210.svg" alt="$\displaystyle f : \mathbb{R}$" loading="lazy"></td>
                    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
                    <td align="LEFT" nowrap="" width="50%"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img212.svg" alt="$\displaystyle \mathbb{R}$" loading="lazy"></td>
                    <td class="eqno" width="10" align="RIGHT">
                    &nbsp;</td></tr>
                    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
                    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
                    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img845.svg" alt="$\displaystyle w = f(u) = {\mathrm{senh}}u$" loading="lazy"></td>
                    <td class="eqno" width="10" align="RIGHT">
                    &nbsp;</td></tr>
                    </tbody></table></div>
                    <br clear="ALL">
                    
                    é sobrejetiva. Além disso, para cada <!-- MATH
                     $w \in \mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img457.svg" alt="$w \in \mathbb{R}$" loading="lazy"></span>, é único o valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> que satisfaz <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img846.svg" alt="$w = f(u)$" loading="lazy"></span>, então esta função é
                    também injetora. Logo, <!-- MATH
                     ${\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span> é uma função bijetora.
                    
                    <p class=" unidade" id="2P100" title="2P100">
                    Conforme <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> aumenta (para o infinito) o tamanho do arco <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img62.svg" alt="$AV$" loading="lazy"></span> também aumenta. Por conseguinte, a ordenada do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>
                    aumenta e o valor de <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img218.svg" alt="$f(u)$" loading="lazy"></span> também aumenta indefinidamente. O mesmo ocorre para os valores negativos de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>. Temos
                    então
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to \infty} {\mathrm{senh}}u = \infty \qquad \text{e} \qquad \lim_{u \to -\infty} {\mathrm{senh}}u = -\infty.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P101" title="2P101">
                    <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">&nbsp; &nbsp;e<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">
                    </div>
                    
                    <p class=" unidade" id="2P102" title="2P102">
                    O gráfico de <!-- MATH
                     $f(u) = {\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img844.svg" alt="$f(u) = {\mathrm{senh}}u$" loading="lazy"></span> é dado por
                    </p>
                    <div class="CENTER"><a id="2271"></a>
                    <table id="2I15" title="2I15">
                    <caption class="BOTTOM"><strong>Figura 2.15:</strong>
                    Gráfico da função seno hiperbólico.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <img src="img/fsinh.png" alt="Image fsinh" loading="lazy"> </div></td></tr>
                    </tbody></table>
                    </div>
                    
                    <p class=" unidade" id="2P103" title="2P103">
                    Podemos notar ainda que é uma função contínua (mostraremos isto formalmente na próxima seção), ímpar e estritamente
                    crescente.
                    </p>
                    <p class=" unidade" id="2P104" title="2P104">
                    Agora a função <!-- MATH
                     $w = f(u) = \cosh u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img850.svg" alt="$w = f(u) = \cosh u$" loading="lazy"></span>.<a name="2273"></a> Para qualquer valor real <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, construímos o arco
                    hiperbólico <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img62.svg" alt="$AV$" loading="lazy"></span> associado ao ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, cujo cosseno hiperbólico é a abscissa do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span>. Notemos que
                    não há nenhuma impossibilidade matemática para o valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e, sendo assim, o domínio da função <!-- MATH
                     $f(u) = \cosh u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img851.svg" alt="$f(u) = \cosh u$" loading="lazy"></span> é o
                    conjunto dos números reais. Fazendo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> variar no conjunto dos números reais, vemos que a abscissa do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> somente
                    poderá assumir valores maiores do que <span class="MATH"><img style="height: 1.66ex; vertical-align: -0.11ex; " src="img/img250.svg" alt="$1$" loading="lazy"></span>, isto é, <!-- MATH
                     $\cosh u \in [1, \infty)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img852.svg" alt="$\cosh u \in [1, \infty)$" loading="lazy"></span>. Isto significa que esta função não é
                    sobrejetora no conjunto dos números reais, mas sim no conjunto <!-- MATH
                     $[1,\infty)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img547.svg" alt="$[1,\infty)$" loading="lazy"></span>. Note também que esta função não é
                    injetora, pois para qualquer valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> temos <!-- MATH
                     $\cosh u = \cosh(-u)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img853.svg" alt="$\cosh u = \cosh(-u)$" loading="lazy"></span>, isto é, é uma função par. Portanto a função
                    </p>
                     <br>
                    <div class="mathdisplay unidade" id="2P105" title="2P105">
                    <!-- MATH
                     \begin{eqnarray*}
                    f: \mathbb{R}& \to & \mathbb{R}\\
                    u & \mapsto & w = f(u) = \cosh u
                    \end{eqnarray*}
                     -->
                    <table cellpadding="0" align="CENTER" width="100%">
                    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img210.svg" alt="$\displaystyle f : \mathbb{R}$" loading="lazy"></td>
                    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
                    <td align="LEFT" nowrap="" width="50%"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img212.svg" alt="$\displaystyle \mathbb{R}$" loading="lazy"></td>
                    <td class="eqno" width="10" align="RIGHT">
                    &nbsp;</td></tr>
                    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
                    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
                    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img854.svg" alt="$\displaystyle w = f(u) = \cosh u$" loading="lazy"></td>
                    <td class="eqno" width="10" align="RIGHT">
                    &nbsp;</td></tr>
                    </tbody></table></div>
                    <br clear="ALL">
                    <p class=" unidade" id="2P106" style="text-indent: 0 !important;" title="2P106">
                    não é bijetora.
                    </p>
                    <p class=" unidade" id="2P107" title="2P107">
                    Conforme o valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> aumenta (para o infinito), o tamanho do arco <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img62.svg" alt="$AV$" loading="lazy"></span> aumenta e a abscissa do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> também
                    aumenta indefinidamente. Uma análise similar para valores negativos de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> levam à mesma conclusão. Temos assim,
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to \infty} \cosh u = \infty \qquad \text{e} \qquad \lim_{u \to -\infty} \cosh u = \infty.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P108" title="2P108">
                    <img style="height: 3.15ex; vertical-align: -1.60ex; " src="img/img855.svg" alt="$\displaystyle \lim_{u \to \infty} \cosh u = \infty$" loading="lazy">&nbsp; &nbsp;e<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">
                    </div>
                    
                    <p class=" unidade" id="2P109" title="2P109">
                    O gráfico desta função é dado por
                    </p>
                    <div class="CENTER"><a id="2281"></a>
                    <table id="2I16" title="2I16">
                    <caption class="BOTTOM"><strong>Figura 2.16:</strong>
                    Gráfico da função cosseno hiperbólico.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <img src="img/fcosh.png" alt="Image fcosh" loading="lazy"> </div></td></tr>
                    </tbody></table>
                    </div>
                    
                    <p class=" unidade" id="2P110" title="2P110">
                    Para a função <!-- MATH
                     $w = f(u) = {\mathrm {tgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img857.svg" alt="$w = f(u) = {\mathrm {tgh}}u$" loading="lazy"></span> vamos usar a identidade <!-- MATH
                     ${\mathrm {tgh}}u = \frac{{\mathrm{senh}}u}{\cosh u}$
                     -->
                    <span class="MATH"><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"></span>.<a name="2285"></a> Fazendo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> variar no conjunto dos números reais, temos apenas que nos preocupar com o denominador,
                    que não pode ser nulo. Como vimos anteriormente, para qualquer valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, temos que <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span> é maior ou igual a 1,
                    e, portanto, o denominador da fração anterior, não se anula. Com isto o domínio da função <!-- MATH
                     $f(u) = {\mathrm {tgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img860.svg" alt="$f(u) = {\mathrm {tgh}}u$" loading="lazy"></span> é todo o
                    conjunto dos números reais.
                    </p>
                    <p class=" unidade" id="2P111" title="2P111">
                    Além disso, como o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> está entre as retas assíntotas <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img708.svg" alt="$y=x$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img709.svg" alt="$y=-x$" loading="lazy"></span>, temos que a abscissa do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> é
                    sempre maior que a ordenada do ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> em módulo, isto é, <!-- MATH
                     $\cosh u > |{\mathrm{senh}}u|$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img861.svg" alt="$\cosh u > \vert{\mathrm{senh}}u\vert$" loading="lazy"></span> para qualquer valor de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>. Isto
                    significa que a fração <!-- MATH
                     ${\mathrm {tgh}}u = \frac{{\mathrm{senh}}u}{\cosh u}$
                     -->
                    <span class="MATH"><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"></span> resultará sempre valores menores que 1 em módulo, isto é,
                    <!-- MATH
                     ${\mathrm {tgh}}u \in (-1,1)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img862.svg" alt="${\mathrm {tgh}}u \in (-1,1)$" loading="lazy"></span>.
                    </p>
                    <p class=" unidade" id="2P112" title="2P112">
                    À medida que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> aumenta indefinidamente, os valores de <!-- MATH
                     ${\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span> tendem a se igualar, pois o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> se
                    aproxima da assíntota <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img708.svg" alt="$y=x$" loading="lazy"></span> e isto significa que quando <!-- MATH
                     $u \to \infty$
                     -->
                    <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img863.svg" alt="$u \to \infty$" loading="lazy"></span> os valores de <!-- MATH
                     ${\mathrm {tgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img864.svg" alt="${\mathrm {tgh}}u$" loading="lazy"></span> se aproximam de 1. No
                    caso em que <!-- MATH
                     $u \to -\infty$
                     -->
                    <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img865.svg" alt="$u \to -\infty$" loading="lazy"></span> então o ponto <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img35.svg" alt="$A$" loading="lazy"></span> se aproxima da assíntota <span class="MATH"><img style="height: 1.82ex; vertical-align: -0.50ex; " src="img/img709.svg" alt="$y=-x$" loading="lazy"></span> e neste caso levamos em conta os sinais
                    de <!-- MATH
                     ${\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span>. Em outras palavras,
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to \infty} {\mathrm {tgh}}u = 1 \qquad \text{e} \qquad \lim_{u \to -\infty} {\mathrm {tgh}}u = -1.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P113" title="2P113">
                    <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">&nbsp; &nbsp;e<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">
                    </div>
                    
                    <p class=" unidade" id="2P114" title="2P114">
                    O gráfico da função <!-- MATH
                     $w = f(u) = {\mathrm {tgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img857.svg" alt="$w = f(u) = {\mathrm {tgh}}u$" loading="lazy"></span> é dado por
                    </p>
                    <div class="CENTER"><a id="2293"></a>
                    <table id="2I17" title="2I17">
                    <caption class="BOTTOM"><strong>Figura 2.17:</strong>
                    Gráfico da função tangente hiperbólica.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <img src="img/ftgh.png" alt="Image ftgh" loading="lazy"> </div></td></tr>
                    </tbody></table>
                    </div>
                    
                    <p class=" unidade" id="2P115" title="2P115">
                    A função <!-- MATH
                     $f(u) = {\mathrm {tgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img860.svg" alt="$f(u) = {\mathrm {tgh}}u$" loading="lazy"></span> é uma função monótona crescente, ímpar, limitada e bijetora de <!-- MATH
                     $\mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span> em <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span>.
                    </p>
                    <p class=" unidade" id="2P116" title="2P116">
                    O estudo da função cotangente hiperbólica, <!-- MATH
                     $w = f(u) = {\mathrm{ctgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img868.svg" alt="$w = f(u) = {\mathrm{ctgh}}u$" loading="lazy"></span>, também será feito analizando a identidade <!-- MATH
                     ${\mathrm{ctgh}}u =
                    \frac{\cosh u}{{\mathrm{senh}}u}$
                     -->
                    <span class="MATH"><img style="height: 2.83ex; vertical-align: -0.85ex; " src="img/img869.svg" alt="${\mathrm{ctgh}}u =
                    \frac{\cosh u}{{\mathrm{senh}}u}$" loading="lazy"></span>.<a name="2297"></a> Para determinar o domínio desta função, como se trata de
                    um quociente, precisamos nos preocupar com o anulamento do denominador. O seno hiperbólico se anula somente no ponto <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img239.svg" alt="$u = 0$" loading="lazy"></span> e, portanto, o domínio de <!-- MATH
                     $f(u) = {\mathrm{ctgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img870.svg" alt="$f(u) = {\mathrm{ctgh}}u$" loading="lazy"></span> é o conjunto <!-- MATH
                     $\mathbb{R}^{*} = \mathbb{R}- \{ 0 \}$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img871.svg" alt="$\mathbb{R}^{*} = \mathbb{R}- \{ 0 \}$" loading="lazy"></span>.
                    </p>
                    <p class=" unidade" id="2P117" title="2P117">
                    Também, como visto anteriormente, o numerador é sempre maior que o denominador, em módulo. Portanto os valores
                    resultantes deste quociente são sempre maiores que 1, em módulo, isto é, a imagem desta função é o conjunto <!-- MATH
                     $(-\infty,
                    -1) \cup (1, \infty)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img619.svg" alt="$(-\infty,-1) \cup (1,\infty)$" loading="lazy"></span>. À medida que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> cresce indefinidamente, os valores de <!-- MATH
                     ${\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span> se aproximam (veja
                    explicação anterior) e, portanto,
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to \infty} {\mathrm{ctgh}}u = 1 \qquad \text{e} \qquad \lim_{u \to -\infty} {\mathrm{ctgh}}u = -1.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P118" title="2P118">
                    <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">&nbsp; &nbsp;e<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">
                    </div>
                    
                    <p class=" unidade" id="2P119" title="2P119">
                    Como o ponto <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img239.svg" alt="$u = 0$" loading="lazy"></span> é um ponto crítico desta função, vamos estudar os limites no ponto 0. A medida que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> se aproxima
                    de <span class="MATH">0</span>, os valores do denomidador <!-- MATH
                     ${\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span>, se aproximam de <span class="MATH">0</span> e a fração vai para o infinito. Temos então, com o
                    respectivo estudo de sinal lateral,
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to 0^{+}} {\mathrm{ctgh}}u = \infty \qquad \text{e} \qquad \lim_{u \to 0^{-}} {\mathrm{ctgh}}u = -\infty.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P120" title="2P120">
                    <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">&nbsp; &nbsp;e<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">
                    </div>
                    
                    <p class=" unidade" id="2P121" title="2P121">
                    O gráfico desta função é dado por
                    </p>
                    <div class="CENTER"><a id="2307"></a>
                    <table id="2I18" title="2I18">
                    <caption class="BOTTOM"><strong>Figura 2.18:</strong>
                    Gráfico da função cotangente hiperbólica.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <img src="img/fctgh.png" alt="Image fctgh" loading="lazy"> </div></td></tr>
                    </tbody></table>
                    </div>
                    
                    <p class=" unidade" id="2P122" title="2P122">
                    A função <!-- MATH
                     $f(u) = {\mathrm{ctgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img870.svg" alt="$f(u) = {\mathrm{ctgh}}u$" loading="lazy"></span> é uma função ímpar e bijetora do conjunto <!-- MATH
                     $\mathbb{R}^{*} = \mathbb{R}-\{0\}$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img871.svg" alt="$\mathbb{R}^{*} = \mathbb{R}- \{ 0 \}$" loading="lazy"></span> no conjunto <!-- MATH
                     $(-\infty, -1) \cup
                    (1, \infty)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img619.svg" alt="$(-\infty,-1) \cup (1,\infty)$" loading="lazy"></span>.
                    </p>
                    <p class=" unidade" id="2P123" title="2P123">
                    Para a função <!-- MATH
                     $w = f(u) = {\mathrm{sech}}u = \frac{1}{\cosh u}$
                     -->
                    <span class="MATH"><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"></span>,<a name="2312"></a> o domínio é o conjunto dos
                    números reais, uma vez que o denominador nunca se anula, mais do que isto, o denominador é sempre maior ou igual a 1.
                    Portanto, os valores assumidos pelo quociente <!-- MATH
                     $\frac{1}{\cosh u}$
                     -->
                    <span class="MATH"><img style="height: 2.83ex; vertical-align: -0.85ex; " src="img/img877.svg" alt="$\frac{1}{\cosh u}$" loading="lazy"></span>, serão sempre positivos e menores ou iguais a <span class="MATH"><img style="height: 1.66ex; vertical-align: -0.11ex; " src="img/img250.svg" alt="$1$" loading="lazy"></span> e,
                    desta forma, a função é limitada inferiormente por 0 e superiormente por 1. Além disso, como cosseno hiperbólico é uma
                    função par, então a função secante também será uma função par.
                    </p>
                    <p class=" unidade" id="2P124" title="2P124">
                    À medida que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> cresce indefinidamente, o denominador também cresce indefinidamente e, portanto,
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to -\infty} {\mathrm{sech}}u = \lim_{u \to \infty} {\mathrm{sech}}u = 0.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P125" title="2P125">
                    <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">
                    </div>
                    
                    <p class=" unidade" id="2P126" title="2P126">
                    O gráfico desta função é
                    </p>
                    <div class="CENTER"><a id="2319"></a>
                    <table id="2I19" title="2I19">
                    <caption class="BOTTOM"><strong>Figura 2.19:</strong>
                    Gráfico da função secante hiperbólica.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <img src="img/fsech.png" alt="Image fsech" loading="lazy"> </div></td></tr>
                    </tbody></table>
                    </div>
                    
                    <p class=" unidade" id="2P127" title="2P127">
                    A cossecante hiperbólica é dada pelo quociente <!-- MATH
                     $w = f(u) = {\mathrm{csch}}u = \frac{1}{{\mathrm{senh}}u}$
                     -->
                    <span class="MATH"><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"></span><a name="2323"></a> e desta forma, seu domínio é o conjunto dos números reais tais que o denominador não se anula, isto
                    é, <!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span>. A imagem por sua vez é também o conjunto <!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span> já que a fração <!-- MATH
                     $\frac{1}{{\mathrm{senh}}u}$
                     -->
                    <span class="MATH"><img style="height: 2.83ex; vertical-align: -0.85ex; " src="img/img881.svg" alt="$\frac{1}{{\mathrm{senh}}u}$" loading="lazy"></span> jamais se anula.
                    Conforme <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> cresce (para o infinito), o denominador também cresce (para o infinito) e, assim,
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to -\infty} {\mathrm{csch}}u = \lim_{u \to \infty} {\mathrm{csch}}u = 0.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P128" title="2P128">
                    <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">
                    </div>
                    
                    <p class=" unidade" id="2P129" title="2P129">
                    Próximo do ponto crítico <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img239.svg" alt="$u = 0$" loading="lazy"></span> os valores do denominador também estarão próximos de <span class="MATH">0</span> e, portanto, fazendo o estudo
                    de sinal, temos
                    <!-- MATH
                     \begin{displaymath}
                    \lim_{u \to 0^{+}} {\mathrm{csch}}u = \infty \qquad \text{e} \qquad \lim_{u \to 0^{-}} {\mathrm{csch}}u = -\infty.
                    \end{displaymath}
                     -->
                    </p>
                    <div class="mathdisplay unidade" id="2P130" title="2P130">
                    <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">&nbsp; &nbsp;e<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">
                    </div>
                    
                    <p class=" unidade" id="2P131" title="2P131">
                    O gráfico da função cossecante hiperbólica é
                    </p>
                    <div class="CENTER"><a id="2335"></a>
                    <table id="2I20" title="2I20">
                    <caption class="BOTTOM"><strong>Figura 2.20:</strong>
                    Gráfico da função cossecante hiperbólica.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <img src="img/fcsch.png" alt="Image fcsch" loading="lazy"> </div></td></tr>
                    </tbody></table>
                    </div>
                    
                    <p class=" unidade" id="2P132" title="2P132">
                    A função cossecante hiperbólica é uma função ímpar bijetora de <!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span> em <!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span>. É decrescente em cada um dos
                    semi-eixos positivo e negativo.
                    </p>
                    <p class=" unidade" id="2P133" title="2P133">
                    A relação completa das funções trigonométricas hiperbólicas, com os domínios e imagens é resumida na próxima tabela.
                    
                    </p>
                    <br>
                    <div class="CENTER"><a id="2358"></a>
                    <table id="2T1" title="2T1">
                    <caption><strong>Tabela 2.1:</strong>
                    Domínio e imagem das funções trigonométricas hiperbólicas.</caption>
                    <tbody><tr><td>
                    <div class="CENTER">
                    <table class="PAD  BORDER">
                    <tbody><tr><td class="LEFT">função</td>
                    <td class="CENTER">domínio</td>
                    <td class="CENTER">imagem</td>
                    </tr>
                    <tr><td class="LEFT"><!-- MATH
                     ${\mathrm{senh}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span></td>
                    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;  <!-- MATH
                     $\mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>  &nbsp;&nbsp;&nbsp;&nbsp;</td>
                    <td class="CENTER"><!-- MATH
                     $\mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
                    </tr>
                    <tr><td class="LEFT"><span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span></td>
                    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;  <!-- MATH
                     $\mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>  &nbsp;&nbsp;&nbsp;&nbsp;</td>
                    <td class="CENTER"><!-- MATH
                     $[1,\infty)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img547.svg" alt="$[1,\infty)$" loading="lazy"></span></td>
                    </tr>
                    <tr><td class="LEFT"><!-- MATH
                     ${\mathrm {tgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img864.svg" alt="${\mathrm {tgh}}u$" loading="lazy"></span></td>
                    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;  <!-- MATH
                     $\mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>  &nbsp;&nbsp;&nbsp;&nbsp;</td>
                    <td class="CENTER"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span></td>
                    </tr>
                    <tr><td class="LEFT"><!-- MATH
                     ${\mathrm{ctgh}}u$
                     -->
                    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img885.svg" alt="${\mathrm{ctgh}}u$" loading="lazy"></span></td>
                    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;  <!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span>  &nbsp;&nbsp;&nbsp;&nbsp;</td>
                    <td class="CENTER"><!-- MATH
                     $(-\infty,-1) \cup (1,\infty)$
                     -->
                    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img619.svg" alt="$(-\infty,-1) \cup (1,\infty)$" loading="lazy"></span></td>
                    </tr>
                    <tr><td class="LEFT"><!-- MATH
                     ${\mathrm{sech}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img886.svg" alt="${\mathrm{sech}}u$" loading="lazy"></span></td>
                    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;  <!-- MATH
                     $\mathbb{R}$
                     -->
                    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>  &nbsp;&nbsp;&nbsp;&nbsp;</td>
                    <td class="CENTER"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img887.svg" alt="$(0,1]$" loading="lazy"></span></td>
                    </tr>
                    <tr><td class="LEFT"><!-- MATH
                     ${\mathrm{csch}}u$
                     -->
                    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img888.svg" alt="${\mathrm{csch}}u$" loading="lazy"></span></td>
                    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;  <!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span>  &nbsp;&nbsp;&nbsp;&nbsp;</td>
                    <td class="CENTER"><!-- MATH
                     $\mathbb{R}^{*}$
                     -->
                    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span></td>
                    </tr>
                    </tbody></table>
                    </div></td></tr>
                    </tbody></table>
                    </div>

:::

## 2.4 Continuidade das funções trigonométricas hiperbólicas {#SECTION00640000000000000000}

::: {.raw_html}
                        
                        <p class=" unidade" id="2P134" title="2P134">
                        Agora vamos mostrar que as funções trigonométricas hiperbólicas são contínuas em cada um dos pontos de definição destas
                        funções. Mais precisamente, mostraremos que
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{x \to a} {\mathrm{senh}}x = {\mathrm{senh}}a \qquad \text{e} \qquad \lim_{x \to a} \cosh x = \cosh a,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P135" title="2P135">
                        <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">&nbsp; &nbsp;e<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">
                        </div><p class=" unidade" id="2P136" style="text-indent: 0 !important;" title="2P136">
                        para qualquer <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span>.
                        
                        </p>
                        
                        <div id="2Teo4" title="2Teo4" class=" unidade"><a id="limfundsinh"><b>Proposição  <span class="arabic">2</span>.<span class="arabic">4</span></b></a> &nbsp; 
                        <i>O limite
                        </i><!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0} \frac{{\mathrm{senh}}u}{u}
                        \end{displaymath}
                         -->
                        
                        <div class="mathdisplay unidade" id="2P137" title="2P137">
                        <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">
                        </div><p class=" unidade" id="2P138" style="text-indent: 0 !important;" title="2P138"><i>
                        existe e é igual a 1.
                        </i></p></div>
                        
                        
                        <div><i>Prova</i>.
                        
                        Vamos estudar os limites laterais e verificar que são ambos iguais a 1. Para obter o limite quando <!-- MATH
                         $u \to 0^{+}$
                         -->
                        <span class="MATH"><img style="height: 1.88ex; vertical-align: -0.13ex; " src="img/img240.svg" alt="$u \to 0^{+}$" loading="lazy"></span>
                        podemos considerar que <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img892.svg" alt="$0 < u < 1$" loading="lazy"></span>. Consideremos o arco hiperbólico <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img62.svg" alt="$AV$" loading="lazy"></span>, relacionado com o ângulo hiperbólico <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e a
                        reta <span class="MATH"><img style="height: 1.52ex; vertical-align: -0.10ex; " src="img/img98.svg" alt="$t$" loading="lazy"></span> paralela ao eixo <span class="MATH"><img style="height: 1.56ex; vertical-align: -0.50ex; " src="img/img71.svg" alt="$y$" loading="lazy"></span> que passa por <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img310.svg" alt="$V$" loading="lazy"></span>. Esta reta intercepta o segmento <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img100.svg" alt="$OA$" loading="lazy"></span> em um ponto que denominaremos <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img6.svg" alt="$Q$" loading="lazy"></span>
                        (Ver figura <a href="#figlimhip">2.21</a>).
                        
                        <div class="CENTER"><a id="figlimhip"></a><a id="2375"></a>
                        <table id="2I21" title="2I21">
                        <caption class="BOTTOM"><strong>Figura 2.21:</strong>
                        Visualização geométrica do limite</caption>
                        <tbody><tr><td>
                        <div class="CENTER">
                        <img src="img/limhip.png" alt="Image limhip" loading="lazy"> </div></td></tr>
                        </tbody></table>
                        </div>
                        
                        <p class=" unidade" id="2P139" title="2P139">
                        Nestes termos, sabemos que a área do setor hiperbólico <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img311.svg" alt="$AOV$" loading="lazy"></span> (a área sombreada da figura (<a href="#figanghippos">2.5</a>)) é igual
                        a <!-- MATH
                         $\frac{u}{2}$
                         -->
                        <span class="MATH"><img style="height: 2.45ex; vertical-align: -0.83ex; " src="img/img61.svg" alt="$\frac{u}{2}$" loading="lazy"></span>, a área do triângulo <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img311.svg" alt="$AOV$" loading="lazy"></span> é igual a <!-- MATH
                         $\frac{{\mathrm{senh}}u}{2}$
                         -->
                        <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img893.svg" alt="$\frac{{\mathrm{senh}}u}{2}$" loading="lazy"></span> e a área do triângulo retângulo <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img894.svg" alt="$OVQ$" loading="lazy"></span> é igual
                        a <!-- MATH
                         $\frac{1}{2} {\mathrm {tgh}}u = \frac{1}{2} \frac{{\mathrm{senh}}u}{\cosh u}$
                         -->
                        <span class="MATH"><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"></span>. Também a área do triângulo <span class="MATH"><img style="height: 2.05ex; vertical-align: -0.50ex; " src="img/img894.svg" alt="$OVQ$" loading="lazy"></span> é menor que a área do
                        setor hiperbólico <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img311.svg" alt="$AOV$" loading="lazy"></span> que por sua vez é menor que a área do triângulo <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img311.svg" alt="$AOV$" loading="lazy"></span>, ou seja,
                        <!-- MATH
                         \begin{displaymath}
                        \frac{{\mathrm{senh}}u}{2 \cosh u} < \frac{u}{2} < \frac{{\mathrm{senh}}u}{2}.
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P140" title="2P140">
                        <img style="height: 4.55ex; vertical-align: -1.57ex; " src="img/img896.svg" alt="$\displaystyle \frac{{\mathrm{senh}}u}{2 \cosh u} < \frac{u}{2} < \frac{{\mathrm{senh}}u}{2}. $" loading="lazy">
                        </div>
                        
                        <p class=" unidade" id="2P141" title="2P141">
                        Multiplicando tudo por 2 e dividindo tudo por <!-- MATH
                         ${\mathrm{senh}}u$
                         -->
                        <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span> (que é positivo), temos
                        <!-- MATH
                         \begin{displaymath}
                        \frac{1}{\cosh u} < \frac{u}{{\mathrm{senh}}u} < 1,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P142" title="2P142">
                        <img style="height: 4.55ex; vertical-align: -1.57ex; " src="img/img897.svg" alt="$\displaystyle \frac{1}{\cosh u} < \frac{u}{{\mathrm{senh}}u} < 1, $" loading="lazy">
                        </div><p class=" unidade" id="2P143" style="text-indent: 0 !important;" title="2P143">
                        ou ainda,
                        <!-- MATH
                         \begin{displaymath}
                        \cosh u > \frac{{\mathrm{senh}}u}{u} > 1.
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P144" title="2P144">
                        <img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img898.svg" alt="$\displaystyle \cosh u > \frac{{\mathrm{senh}}u}{u} > 1. $" loading="lazy">
                        </div>
                        
                        <p class=" unidade" id="2P145" title="2P145">
                        Da primeira desigualdade temos que <!-- MATH
                         ${\mathrm{senh}}u < u \cosh u$
                         -->
                        <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img899.svg" alt="${\mathrm{senh}}u < u \cosh u$" loading="lazy"></span> e, usando isto, temos
                        </p>
                        <div class="mathdisplay unidade" id="2P146" title="2P146"><table class="equation">
                        <tbody><tr>
                        <td style="text-align:right;"><span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img900.svg" alt="$\displaystyle \cosh u$" loading="lazy"></span></td>
                        <td style="text-align:left;"><span class="MATH"><img style="height: 2.80ex; vertical-align: -0.29ex; " src="img/img901.svg" alt="$\displaystyle = \sqrt{1+{\mathrm{senh}}^{2} u}$" loading="lazy"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><img style="height: 2.80ex; vertical-align: -0.29ex; " src="img/img902.svg" alt="$\displaystyle < \sqrt{1+2{\mathrm{senh}}u + {\mathrm{senh}}^{2}u}$" loading="lazy"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><img style="height: 1.83ex; vertical-align: -0.27ex; " src="img/img903.svg" alt="$\displaystyle = 1 + {\mathrm{senh}}u < 1 + u \cosh u.$" loading="lazy"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        </tbody></table></div>
                        
                        
                        <p class=" unidade" id="2P147" title="2P147">
                        Desta forma <!-- MATH
                         $\cosh u < 1 + u \cosh u$
                         -->
                        <span class="MATH"><img style="height: 1.83ex; vertical-align: -0.27ex; " src="img/img904.svg" alt="$\cosh u < 1 + u \cosh u$" loading="lazy"></span> e, reorganizando os termos, temos
                        <!-- MATH
                         \begin{displaymath}
                        (1-u) \cosh u < 1,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P148" title="2P148">
                        <img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img905.svg" alt="$\displaystyle (1-u) \cosh u < 1, $" loading="lazy">
                        </div><p class=" unidade" id="2P149" style="text-indent: 0 !important;" title="2P149">
                        e como <!-- MATH
                         $u \in (0,1)$
                         -->
                        <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img906.svg" alt="$u \in (0,1)$" loading="lazy"></span> então <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img907.svg" alt="$(1-u) > 0$" loading="lazy"></span>, o que nos permite obter <!-- MATH
                         $\cosh u < \frac{1}{1-u}$
                         -->
                        <span class="MATH"><img style="height: 2.94ex; vertical-align: -0.95ex; " src="img/img908.svg" alt="$\cosh u < \frac{1}{1-u}$" loading="lazy"></span>. Segue que
                        <!-- MATH
                         \begin{displaymath}
                        1 < \frac{{\mathrm{senh}}u}{u} < \cosh u < \frac{1}{1-u}.
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P150" title="2P150">
                        <img style="height: 4.69ex; vertical-align: -1.72ex; " src="img/img909.svg" alt="$\displaystyle 1 < \frac{{\mathrm{senh}}u}{u} < \cosh u < \frac{1}{1-u}. $" loading="lazy">
                        </div>
                        
                        <p class=" unidade" id="2P151" title="2P151">
                        Passando agora o limite na desigualdade, quando <!-- MATH
                         $u \to 0^{+}$
                         -->
                        <span class="MATH"><img style="height: 1.88ex; vertical-align: -0.13ex; " src="img/img240.svg" alt="$u \to 0^{+}$" loading="lazy"></span>, temos que o limite do termo do lado esquerdo existe e é
                        igual a 1 e o limite do lado direito também existe e é igual a 1, pois <!-- MATH
                         $\frac{1}{1-u}$
                         -->
                        <span class="MATH"><img style="height: 2.94ex; vertical-align: -0.95ex; " src="img/img910.svg" alt="$\frac{1}{1-u}$" loading="lazy"></span> é uma função contínua em <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img239.svg" alt="$u = 0$" loading="lazy"></span>. Temos então pelo teorema do confronto (teorema do sanduíche) que
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0^{+}} \frac{{\mathrm{senh}}u}{u} = 1.
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P152" title="2P152">
                        <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">
                        </div>
                        
                        <p class=" unidade" id="2P153" title="2P153">
                        O caso em que <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img52.svg" alt="$u < 0$" loading="lazy"></span>, é obtido observando que a função <!-- MATH
                         $\frac{{\mathrm{senh}}u}{u}$
                         -->
                        <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img912.svg" alt="$\frac{{\mathrm{senh}}u}{u}$" loading="lazy"></span> é uma função par. Assim o comportamento à
                        esquerda de 0 é o mesmo comportamento à direita de zero. Temos então
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0^{-}} \frac{{\mathrm{senh}}u}{u} = \lim_{u \to 0^{+}} \frac{{\mathrm{senh}}u}{u} = 1,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P154" title="2P154">
                        <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, $" loading="lazy">
                        </div><p class=" unidade" id="2P155" style="text-indent: 0 !important;" title="2P155">
                        e, portanto,
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0} \frac{{\mathrm{senh}}u}{u} = 1,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P156" title="2P156">
                        <img style="height: 4.75ex; vertical-align: -1.78ex; " src="img/img914.svg" alt="$\displaystyle \lim_{u \to 0} \frac{{\mathrm{senh}}u}{u} = 1, $" loading="lazy">
                        </div>
                        <p style="text-indent: 0 !important; text-align: left !important;" title="2P156"><span class="MATH">o que encerra esta demonstração.</span><span style="text-align: right;"><img style="height: 1.59ex; vertical-align: -0.10ex; float: right;" src="img/img193.svg" alt="$\qedsymbol$" loading="lazy">
                        </span></p>
                        </div>
                        
                        
                        <div id="2Teo5" title="2Teo5" class=" unidade"><a id="teo2.5"><b>Teorema  <span class="arabic">2</span>.<span class="arabic">5</span></b></a> &nbsp; 
                        <i>As funções seno e cosseno hiperbólico são contínuas em <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img239.svg" alt="$u = 0$" loading="lazy"></span>, isto é,
                        </i><!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0} \cosh u = 1 \qquad \text{e} \qquad \lim_{u \to 0} {\mathrm{senh}}u = 0.
                        \end{displaymath}
                         -->
                        
                        <div class="mathdisplay unidade" id="2P157" title="2P157">
                        <img style="height: 3.33ex; vertical-align: -1.78ex; " src="img/img915.svg" alt="$\displaystyle \lim_{u \to 0} \cosh u = 1$" loading="lazy">&nbsp; &nbsp;e<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">
                        </div></div>
                        
                        
                        <div><i>Prova</i>.
                        
                        Para o primeiro limite, supondo primeiro <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img306.svg" alt="$u>0$" loading="lazy"></span> (mais precisamente <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img892.svg" alt="$0 < u < 1$" loading="lazy"></span>), usamos a desigualdade obtida no teorema
                        anterior
                        <!-- MATH
                         \begin{displaymath}
                        1 < \frac{{\mathrm{senh}}u}{u} < \cosh u < \frac{1}{1-u},
                        \end{displaymath}
                         -->
                        
                        <div class="mathdisplay unidade" id="2P158" title="2P158">
                        <img style="height: 4.69ex; vertical-align: -1.72ex; " src="img/img917.svg" alt="$\displaystyle 1 < \frac{{\mathrm{senh}}u}{u} < \cosh u < \frac{1}{1-u}, $" loading="lazy">
                        </div><p class=" unidade" id="2P159" style="text-indent: 0 !important;" title="2P159">
                        e o teorema do confronto garante que <!-- MATH
                         $\lim\limits_{u \to 0^{+}} \cosh u = 1$
                         -->
                        <span class="MATH"><img style="height: 3.37ex; vertical-align: -1.82ex; " src="img/img918.svg" alt="$\lim\limits_{u \to 0^{+}} \cosh u = 1$" loading="lazy"></span>. Para <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img52.svg" alt="$u < 0$" loading="lazy"></span> lembremos que cosseno
                        hiperbólico é uma função par e então como na demonstração do teorema anterior,
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0^{-}} \cosh u = \lim_{u \to 0^{+}} \cosh u = 1,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P160" title="2P160">
                        <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">
                        </div><p class=" unidade" id="2P161" style="text-indent: 0 !important;" title="2P161">
                        e isso prova o primeiro limite.
                        </p>
                        <p class=" unidade" id="2P162" title="2P162">
                        Para provar o segundo limite, usaremos o item (c) do teorema <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teopop">1.2</a>. Como os limites de <!-- MATH
                         $\frac{{\mathrm{senh}}u}{u}$
                         -->
                        <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img912.svg" alt="$\frac{{\mathrm{senh}}u}{u}$" loading="lazy"></span> e de
                        <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> existem quando <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img270.svg" alt="$u \to 0$" loading="lazy"></span> então o limite do produto existe e
                        <!-- MATH
                         \begin{displaymath}
                        \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.
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P163" title="2P163">
                        <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...
                        ...m_{u \to 0} u \cdot \lim_{u \to 0} \frac{{\mathrm{senh}}u}{u} = 0 \cdot 1 = 0. $" loading="lazy">
                        </div>
                        
                        <p class=" unidade" id="2P164" title="2P164">
                        Agora, como <!-- MATH
                         $\frac{u {\mathrm{senh}}u}{u} = {\mathrm{senh}}u$
                         -->
                        <span class="MATH"><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"></span> para todo <span class="MATH"><img style="height: 2.06ex; vertical-align: -0.51ex; " src="img/img339.svg" alt="$u \neq 0$" loading="lazy"></span> então do teorema <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teofgdif">1.4</a> segue que
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0} {\mathrm{senh}}u = \lim_{u \to 0} \frac{u {\mathrm{senh}}u}{u} = 0,
                        \end{displaymath}
                         -->
                        </p>
                        <div class="mathdisplay unidade" id="2P165" title="2P165">
                        <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, $" loading="lazy">
                        </div>
                        <p style="text-indent: 0 !important; text-align: left !important;" title="2P165"><span class="MATH">e isso finaliza esta demonstração.</span><span style="text-align: right;"><img style="height: 1.59ex; vertical-align: -0.10ex; float: right;" src="img/img193.svg" alt="$\qedsymbol$" loading="lazy">
                        </span></p>
                        </div>
                        
                       
                        <div id="2Teo6" title="2Teo6" class=" unidade"><a id="teo2.6"><b>Teorema  <span class="arabic">2</span>.<span class="arabic">6</span></b></a> &nbsp; 
                        <i>Para qualquer <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span> tem-se
                        </i><!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0} \cosh(u+a) = \cosh a \qquad \text{e} \qquad \lim_{u \to 0} {\mathrm{senh}}(u+a) = {\mathrm{senh}}a.
                        \end{displaymath}
                         -->
                        
                        <div class="mathdisplay unidade" id="2P166" title="2P166">
                        <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">&nbsp; &nbsp;e<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">
                        </div></div>
                        
                        
                        <div><i>Prova</i>.
                        
                        Usando a identidade trigonométrica para a soma de arcos do cosseno hiperbólico, temos que
                        <!-- MATH
                         \begin{displaymath}
                        \lim_{u \to 0} \cosh(u+a) = \lim_{u \to 0} [\cosh u \cosh a + {\mathrm{senh}}u {\mathrm{senh}}a],
                        \end{displaymath}
                         -->
                        
                        <div class="mathdisplay unidade" id="2P167" title="2P167">
                        <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">
                        </div>
                        e dos itens (a) e (b) do teorema <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teopop">1.2</a>, segue que
                        
                        <div class="mathdisplay unidade" id="2P168" title="2P168"><table class="equation">
                        <tbody><tr>
                        <td style="text-align:right;"><span class="MATH"><img style="height: 3.45ex; vertical-align: -1.78ex; " src="img/img926.svg" alt="$\displaystyle \lim_{u \to 0} \cosh(u+a)$" loading="lazy"></span></td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        </tbody></table></div>
                        
                        
                        <p class=" unidade" id="2P169" title="2P169">
                        Usando agora a identidade trigonométrica para a soma de arcos do seno hiperbólico, temos
                        </p>
                        <div class="mathdisplay unidade" id="2P170" title="2P170"><table class="equation">
                        <tbody><tr>
                        <td style="text-align:right;"><span class="MATH"><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"></span></td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        <tr>
                        <td>&nbsp;</td>
                        <td style="text-align:left;"><span class="MATH"><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"></span></td>
                        <td class="eqno" style="text-align:right">
                        &nbsp;&nbsp;&nbsp;</td></tr>
                        </tbody></table></div>
                        
                        <p style="text-indent: 0 !important; text-align: left !important;" title="2P170"><span class="MATH">e isso termina esta demonstração.</span><span style="text-align: right;"><img style="height: 1.59ex; vertical-align: -0.10ex; float: right;" src="img/img193.svg" alt="$\qedsymbol$" loading="lazy">
                        </span></p>
                        </div>
                        
                        <p class=" unidade" id="2P171" title="2P171">
                        Os limites indicados no início desta seção seguem agora imediatamente do teorema de mudança de variáveis <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teomudv">1.5</a>,
                        e dos limites que acabamos de provar.
                        </p>
                      
                        <div id="2Teo7" title="2Teo7" class=" unidade"><a id="cor2.7"><b>Corolário  <span class="arabic">2</span>.<span class="arabic">7</span></b></a> &nbsp; 
                        <i>As funções seno e cosseno hiperbólicos são contínuas em qualquer ponto <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span>, isto é,
                        </i><!-- MATH
                         \begin{displaymath}
                        \lim_{u \to a} \cosh u = \cosh a \qquad \text{e} \qquad \lim_{u \to a} {\mathrm{senh}}u = {\mathrm{senh}}a.
                        \end{displaymath}
                         -->
                        
                        <div class="mathdisplay unidade" id="2P172" title="2P172">
                        <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">&nbsp; &nbsp;e<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">
                        </div></div>
                        
                        <p class=" unidade" id="2P173" title="2P173">
                        Vamos agora analisar a continuidade das outras quatro funções trigonométricas hiperbólicas, já que estas são escritas
                        como um quociente em termos de seno e cosseno. Usando o item (d) do teorema <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teopop">1.2</a>, podemos facilmente provar as
                        afirmações a seguir.
                        </p>
                        <p class=" unidade" id="2P174" title="2P174">
                        As funções tangente, cotangente, secante e cossecante hiperbólicas são contínuas nos seus domínios de definição. Isto
                        é,
                        </p>
                        <div class="CENTER">
                        <table class="PAD ">
                        <tbody><tr><td class="CENTER">&nbsp;</td>
                        <td class="LEFT"><!-- MATH
                         $\lim\limits_{u \to a} {\mathrm {tgh}}u = {\mathrm {tgh}}a$
                         -->
                        <span class="MATH"><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"></span>, para todo <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span>,</td>
                        </tr>
                        <tr><td class="CENTER">&nbsp;</td>
                        <td class="LEFT"><!-- MATH
                         $\lim\limits_{u \to a} {\mathrm{ctgh}}u = {\mathrm{ctgh}}a$
                         -->
                        <span class="MATH"><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"></span>, para todo <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span> com <span class="MATH"><img style="height: 2.06ex; vertical-align: -0.51ex; " src="img/img939.svg" alt="$a \neq 0$" loading="lazy"></span>,</td>
                        </tr>
                        <tr><td class="CENTER">&nbsp;</td>
                        <td class="LEFT"><!-- MATH
                         $\lim\limits_{u \to a} {\mathrm{sech}}u = {\mathrm{sech}}a$
                         -->
                        <span class="MATH"><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"></span>, para todo <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span> e</td>
                        </tr>
                        <tr><td class="CENTER">&nbsp;</td>
                        <td class="LEFT"><!-- MATH
                         $\lim\limits_{u \to a} {\mathrm{csch}}u = {\mathrm{csch}}a$
                         -->
                        <span class="MATH"><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"></span>, para todo <!-- MATH
                         $a \in \mathbb{R}$
                         -->
                        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span> com <span class="MATH"><img style="height: 2.06ex; vertical-align: -0.51ex; " src="img/img939.svg" alt="$a \neq 0$" loading="lazy"></span>.</td>
                        </tr>
                        </tbody></table>
                        </div>

:::

## 2.5 Derivadas de funções trigonométricas hiperbólicas {#SECTION00650000000000000000}

::: {.raw_html}

                       <a id="secderhip"></a>
                         
                            
                            <p class=" unidade" id="2P175" title="2P175">
                            Vamos agora deduzir as derivadas das funções trigonométricas hiperbólicas. Para isto usaremos primeiro a definição de
                            derivada, isto é,
                            <!-- MATH
                             \begin{displaymath}
                            f'(u) = \lim_{h \to 0} \frac{f(u+h) - f(u)}{h}
                            \end{displaymath}
                             -->
                            </p>
                            <div class="mathdisplay unidade" id="2P176" title="2P176">
                            <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">
                            </div><p class=" unidade" id="2P177" title="2P177">
                            para encontrar as derivadas de <!-- MATH
                             ${\mathrm{senh}}u$
                             -->
                            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span>. Depois usaremos a regra do quociente para obter as derivadas das
                            demais funções trigonométricas hiperbólicas. Antes precisamos determinar um limite importante.
                            
                             </p>
                            
                            <div id="2Teo8" title="2Teo8" class=" unidade"><a id="prop2.8"><b>Proposição  <span class="arabic">2</span>.<span class="arabic">8</span></b></a> &nbsp; 
                            <i>O limite
                            </i><!-- MATH
                             \begin{displaymath}
                            \lim_{h \to 0} \frac{\cosh h - 1}{h}
                            \end{displaymath}
                             -->
                            
                            <div class="mathdisplay unidade" id="2P178" title="2P178">
                            <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">
                            </div><i>
                            existe e é igual a 0.
                            </i></div>
                            
                            
                            <div><i>Prova</i>.
                            
                            Vamos modificar um pouco o quociente deste limite e usar a proposição anterior. Notemos que
                        <!-- MATH
                             \begin{displaymath}
                            \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)}.
                            \end{displaymath}
                             -->
                        
                            <div class="mathdisplay unidade" id="2P179" title="2P179">
                            <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">
                            </div>
                            
                            <p class=" unidade" id="2P180" title="2P180">
                            Usando agora a identidade fundamental (<a href="#idfundhip">2.3</a>) no membro da direita, temos que
                            <!-- MATH
                             \begin{displaymath}
                            \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}.
                            \end{displaymath}
                             -->
                            </p>
                            <div class="mathdisplay unidade" id="2P181" title="2P181">
                            <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">
                            </div>
                            
                            <p class=" unidade" id="2P182" title="2P182">
                            Olhando para o membro da direira, temos que, o limite da primeira fração quando <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img946.svg" alt="$h \to 0$" loading="lazy"></span> existe e é igual a 1
                            (proposição <a href="#limfundsinh">2.4</a>) e o limite da segunda fração quando <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img946.svg" alt="$h \to 0$" loading="lazy"></span> também existe por ser uma função contínua
                            em <span class="MATH"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img293.svg" alt="$h$" loading="lazy"></span>. Desta forma o limite do produto existe quando <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img946.svg" alt="$h \to 0$" loading="lazy"></span> e,
                            </p>
                            <div class="mathdisplay unidade" id="2P183" title="2P183"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><img style="height: 4.75ex; vertical-align: -1.78ex; " src="img/img943.svg" alt="$\displaystyle \lim_{h \to 0} \frac{\cosh h - 1}{h} $" loading="lazy"></span></td>
                            <td style="text-align:left;"><span class="MATH"><img style="height: 4.75ex; vertical-align: -1.78ex; " src="img/img947.svg" alt="$\displaystyle = \lim_{h \to 0} \frac{{\mathrm{senh}}h}{h} \cdot \frac{{\mathrm{senh}}h}{\cosh h + 1}$" loading="lazy"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><img style="height: 5.24ex; vertical-align: -2.10ex; " src="img/img948.svg" alt="$\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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            
                            <p style="text-indent: 0 !important; text-align: left !important;" title="2P183"><span class="MATH">e a prova está terminada.</span><span style="text-align: right;"><img style="height: 1.59ex; vertical-align: -0.10ex; float: right;" src="img/img193.svg" alt="$\qedsymbol$" loading="lazy">
                            </span></p>
                            </div>
                            
                            <p class=" unidade" id="2P184" title="2P184">
                            Agora temos condições de deduzir as fórmulas de derivada para as funções trigonométricas seno e cosseno
                            hiperbólicos. Para a função seno hiperbólico temos que a derivada é dada por, <a name="2552"></a>
                            <!-- MATH
                             \begin{displaymath}
                            \frac{d}{du} {\mathrm{senh}}u = \lim_{h \to 0} \frac{{\mathrm{senh}}(u+h) - {\mathrm{senh}}u}{h}
                            \end{displaymath}
                             -->
                            </p>
                            <div class="mathdisplay unidade" id="2P185" title="2P185">
                            <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">
                            </div><p class=" unidade" id="2P186" style="text-indent: 0 !important;" title="2P186">
                            em todos os valores <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span> tais que o limite existe.
                            </p>
                            <p class=" unidade" id="2P187" title="2P187">
                            Assim,
                            </p>
                            <div class="mathdisplay unidade" id="2P188" title="2P188"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img950.svg" alt="$\displaystyle \frac{d}{du} {\mathrm{senh}}u$" loading="lazy"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            <p class=" unidade" id="2P189" style="text-indent: 0 !important;" title="2P189">
                            para todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span> tal que o limite acima existe.
                            </p>
                            <p class=" unidade" id="2P190" title="2P190">
                            Mas os limites de <!-- MATH
                             ${\mathrm{senh}}u \frac{(\cosh h - 1)}{h}$
                             -->
                            <span class="MATH"><img style="height: 3.11ex; vertical-align: -0.83ex; " src="img/img954.svg" alt="${\mathrm{senh}}u \frac{(\cosh h - 1)}{h}$" loading="lazy"></span> e <!-- MATH
                             $\frac{{\mathrm{senh}}h}{h} \cosh u$
                             -->
                            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img955.svg" alt="$\frac{{\mathrm{senh}}h}{h} \cosh u$" loading="lazy"></span> existem para todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span> e assim,
                            </p>
                            <div class="mathdisplay unidade" id="2P191" title="2P191"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img950.svg" alt="$\displaystyle \frac{d}{du} {\mathrm{senh}}u$" loading="lazy"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><img style="height: 5.24ex; vertical-align: -2.10ex; " src="img/img956.svg" alt="$\displaystyle = {\mathrm{senh}}u \left( \lim_{h \to 0} \frac{\cosh h - 1}{h} \r...
                            ...) + \cosh u \left( \lim_{h \to 0} \frac{{\mathrm{senh}}h}{h} \right)
                            = \cosh u,$" loading="lazy"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            <p class=" unidade" id="2P192" style="text-indent: 0 !important;" title="2P192">
                            para todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>.
                            </p>
                            <p class=" unidade" id="2P193" title="2P193">
                            Para a função cosseno hiperbólico, temos que <a name="2592"></a>
                            <!-- MATH
                             \begin{displaymath}
                            \frac{d}{du} \cosh u = \lim_{h \to 0} \frac{\cosh(u+h) - \cosh u}{h}
                            \end{displaymath}
                             -->
                            </p>
                            <div class="mathdisplay unidade" id="2P194" title="2P194">
                            <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">
                            </div><p class=" unidade" id="2P195" style="text-indent: 0 !important;" title="2P195">
                            para todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span> tal que o limite exista. Para tais <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, temos
                            </p>
                            <div class="mathdisplay unidade" id="2P196" title="2P196"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img958.svg" alt="$\displaystyle \frac{d}{du} \cosh u$" loading="lazy"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            
                            
                            <p class=" unidade" id="2P197" title="2P197">
                            Como os limites de cada uma das frações <!-- MATH
                             $\cosh u \frac{(\cosh h - 1)}{h}$
                             -->
                            <span class="MATH"><img style="height: 3.11ex; vertical-align: -0.83ex; " src="img/img962.svg" alt="$\cosh u \frac{(\cosh h - 1)}{h}$" loading="lazy"></span> e <!-- MATH
                             $\frac{{\mathrm{senh}}h}{h} {\mathrm{senh}}u$
                             -->
                            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img963.svg" alt="$\frac{{\mathrm{senh}}h}{h} {\mathrm{senh}}u$" loading="lazy"></span> existem para
                            todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span> então
                            </p>
                            <div class="mathdisplay unidade" id="2P198" title="2P198"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><img style="height: 4.52ex; vertical-align: -1.55ex; " src="img/img958.svg" alt="$\displaystyle \frac{d}{du} \cosh u$" loading="lazy"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><img style="height: 5.24ex; vertical-align: -2.10ex; " src="img/img965.svg" alt="$\displaystyle = \cosh u \left( \lim_{h \to 0} \frac{\cosh h - 1}{h} \right)+ {\...
                            ...}u \left( \lim_{h \to 0} \frac{{\mathrm{senh}}h}{h} \right) =
                            {\mathrm{senh}}u,$" loading="lazy"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            <p class=" unidade" id="2P199" style="text-indent: 0 !important;" title="2P199">
                            para todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>.
                            </p>
                            <p class=" unidade" id="2P200" title="2P200">
                            Para as demais funções trigonométricas hiperbólicas usaremos as identidades em termos de seno e cosseno hiperbólico e a
                            regra de derivação do quociente. Já que as funções seno e cosseno hiperólico são diferenciáveis em todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>
                            então os quocientes de definição das demais funções trigonométricas hiperbólicas são diferenciáveis em todos os pontos
                            onde o denominador não se anula.
                            </p>
                            <p class=" unidade" id="2P201" title="2P201">
                            A função <!-- MATH
                             $f(u) = {\mathrm {tgh}}u$
                             -->
                            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img860.svg" alt="$f(u) = {\mathrm {tgh}}u$" loading="lazy"></span> é diferenciável em todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span> e <a name="2632"></a>
                            </p>
                            <div class="mathdisplay unidade" id="2P202" title="2P202"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><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"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            
                            
                            <p class=" unidade" id="2P203" title="2P203">
                            Para a função cotangente, temos em todo <!-- MATH
                             $u \in \mathbb{R}^{*}$
                             -->
                            <span class="MATH"><img style="height: 1.75ex; vertical-align: -0.18ex; " src="img/img969.svg" alt="$u \in \mathbb{R}^{*}$" loading="lazy"></span>, <a name="2649"></a>
                            </p>
                            <div class="mathdisplay unidade" id="2P204" title="2P204"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><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"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            
                            
                            <p class=" unidade" id="2P205" title="2P205">
                            E, finalmente, para todo <!-- MATH
                             $u \in \mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>, <a name="2665"></a>
                            </p>
                            <div class="mathdisplay unidade" id="2P206" title="2P206"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><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"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            <p class=" unidade" id="2P207" style="text-indent: 0 !important;" title="2P207">
                            e também, para todo <!-- MATH
                             $u \in \mathbb{R}^{*}$
                             -->
                            <span class="MATH"><img style="height: 1.75ex; vertical-align: -0.18ex; " src="img/img969.svg" alt="$u \in \mathbb{R}^{*}$" loading="lazy"></span>, <a name="2683"></a>
                            </p>
                            <div class="mathdisplay unidade" id="2P208" title="2P208"><table class="equation">
                            <tbody><tr>
                            <td style="text-align:right;"><span class="MATH"><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"></span></td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            <tr>
                            <td>&nbsp;</td>
                            <td style="text-align:left;"><span class="MATH"><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"></span></td>
                            <td class="eqno" style="text-align:right">
                            &nbsp;&nbsp;&nbsp;</td></tr>
                            </tbody></table></div>
                            
                            
                            <p class=" unidade" id="2P209" title="2P209">
                            A tabela abaixo, reúne as fórmulas de derivação para as funções trigonométricas hiperbólicas. O conjunto domínio
                            descrito na tabela é o domínio da derivada. Note a semelhança com o caso circular.
                            
                            </p>
                            <br>
                            <div class="CENTER"><a id="2710"></a>
                            <table id="2T2" title="2T2">
                            <caption><strong>Tabela 2.2:</strong>
                            Derivadas das funções trigonométricas hiperbólicas.</caption>
                            <tbody><tr><td>
                            <div class="CENTER">
                            <table class="PAD  BORDER">
                            <tbody><tr><td class="LEFT">função</td>
                            <td class="CENTER">domínio</td>
                            <td class="CENTER">derivada</td>
                            </tr>
                            <tr><td class="LEFT"><!-- MATH
                             ${\mathrm{senh}}u$
                             -->
                            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $\mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
                            <td class="CENTER"><span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span></td>
                            </tr>
                            <tr><td class="LEFT"><span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img859.svg" alt="$\cosh u$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $\mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             ${\mathrm{senh}}u$
                             -->
                            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img847.svg" alt="${\mathrm{senh}}u$" loading="lazy"></span></td>
                            </tr>
                            <tr><td class="LEFT"><!-- MATH
                             ${\mathrm {tgh}}u$
                             -->
                            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img864.svg" alt="${\mathrm {tgh}}u$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $\mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             ${\mathrm{sech}}^{2} u$
                             -->
                            <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img979.svg" alt="${\mathrm{sech}}^{2} u$" loading="lazy"></span></td>
                            </tr>
                            <tr><td class="LEFT"><!-- MATH
                             ${\mathrm{ctgh}}u$
                             -->
                            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img885.svg" alt="${\mathrm{ctgh}}u$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $\mathbb{R}^{*}$
                             -->
                            <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $-{\mathrm{csch}}^{2} u$
                             -->
                            <span class="MATH"><img style="height: 2.28ex; vertical-align: -0.27ex; " src="img/img980.svg" alt="$-{\mathrm{csch}}^{2} u$" loading="lazy"></span></td>
                            </tr>
                            <tr><td class="LEFT"><!-- MATH
                             ${\mathrm{sech}}u$
                             -->
                            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img886.svg" alt="${\mathrm{sech}}u$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $\mathbb{R}$
                             -->
                            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
                            <td class="CENTER"><!-- MATH
                             $-{\mathrm{sech}}u {\mathrm {tgh}}u$
                             -->
                            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img981.svg" alt="$-{\mathrm{sech}}u {\mathrm {tgh}}u$" loading="lazy"></span></td>
                            </tr>
                            <tr><td class="LEFT"><!-- MATH
                             ${\mathrm{csch}}u$
                             -->
                            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img888.svg" alt="${\mathrm{csch}}u$" loading="lazy"></span></td>
                            <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<!-- MATH
                             $\mathbb{R}^{*}$
                             -->
                            <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
                            <td class="CENTER"><!-- MATH
                             $-{\mathrm{csch}}u {\mathrm{ctgh}}u$
                             -->
                            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img982.svg" alt="$-{\mathrm{csch}}u {\mathrm{ctgh}}u$" loading="lazy"></span></td>
                            </tr>
                            </tbody></table>
                            </div></td></tr>
                            </tbody></table>
                            </div>

:::

## 2.6 Funções trigonométricas hiperbólicas inversas {#SECTION00660000000000000000}

::: {.raw_html}
    
    <p class=" unidade" id="2P210" title="2P210">
    Nesta seção, vamos definir as funções trigonométricas inversas, estabelecendo os domínios, as imagens e indicando
    alguns limites importantes. Também apresentaremos os gráficos destas funções. Este não é um trabalho muito fácil pois,
    como acabamos de ver, as funções trigonométricas hiperbólicas não são todas elas bijetoras. Já passamos por este
    problema na seção <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#secfcinv">1.5</a> com as funções trigonométricas circulares. Vamos impor, quando necessário, condições
    de restrição de domínio e de imagem para tornar as funções bijetivas.
    </p>
    <p class=" unidade" id="2P211" title="2P211">
    Comecemos com a função seno hiperbólico, que como vimos anteriormente, é uma função bijetora de <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span> em <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>. Desta
    forma, podemos obter a função inversa do seno hiperbólico, para qualquer valor real. Dado <!-- MATH
     $u \in \mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>, o seno
    hiperbólico inverso de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, é o número <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span>, representado por <!-- MATH
     $w = {\mathrm{senh}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img983.svg" alt="$w = {\mathrm{senh}}^{-1} u$" loading="lazy"></span>, que satisfaz <!-- MATH
     $u = {\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img984.svg" alt="$u = {\mathrm{senh}}w$" loading="lazy"></span>. É usual
    representar também a função seno hiperbólico inverso por <!-- MATH
     $w = \operatorname{arcsinh} u$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img985.svg" alt="$w = \operatorname{arcsinh} u$" loading="lazy"></span> e lemos “arco seno
    hiperbólico”. Vamos usar neste texto a primeira notação e lembre-se de não confundir <!-- MATH
     ${\mathrm{senh}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img986.svg" alt="${\mathrm{senh}}^{-1} u$" loading="lazy"></span> com <!-- MATH
     $({\mathrm{senh}}
    u)^{-1}$
     -->
    <span class="MATH"><img style="height: 2.55ex; vertical-align: -0.62ex; " src="img/img987.svg" alt="$({\mathrm{senh}}
    u)^{-1}$" loading="lazy"></span>. A segunda expressão é o inverso multiplicativo do seno hiperbólico, ou seja a cossecante hiperbólica.
    </p>
    <p class=" unidade" id="2P212" title="2P212">
    Fazendo <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> variar em <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>, temos a função seno hiperbólico inverso, <a name="2718"></a>
    </p>
    <div class="mathdisplay unidade" id="2P213" title="2P213">
    <!-- MATH
     \begin{eqnarray*}
    f : \mathbb{R}& \to & \mathbb{R}\\
    u & \mapsto & w = f(u) = {\mathrm{senh}}^{-1} u,
    \end{eqnarray*}
     -->
    <table cellpadding="0" align="CENTER" width="100%">
    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.65ex; vertical-align: -0.10ex; " src="img/img210.svg" alt="$\displaystyle f : \mathbb{R}$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img212.svg" alt="$\displaystyle \mathbb{R}$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img988.svg" alt="$\displaystyle w = f(u) = {\mathrm{senh}}^{-1} u,$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    </tbody></table></div>
    <br clear="ALL">
    <p class=" unidade" id="2P214" style="text-indent: 0 !important;" title="2P214">
    que satisfaz a relação <!-- MATH
     $u = {\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img984.svg" alt="$u = {\mathrm{senh}}w$" loading="lazy"></span>. Se fizermos <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> tender para o infinito, a relação <!-- MATH
     $u = {\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img984.svg" alt="$u = {\mathrm{senh}}w$" loading="lazy"></span> nos diz que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span>
    também deve ir para o infinito e analogamente para <!-- MATH
     $u \to -\infty$
     -->
    <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img865.svg" alt="$u \to -\infty$" loading="lazy"></span>. Temos assim,
    <!-- MATH
     \begin{displaymath}
    \lim_{u \to \infty} {\mathrm{senh}}^{-1} u = \infty \qquad \text{e} \qquad \lim_{u \to -\infty} {\mathrm{senh}}^{-1} u = -\infty.
    \end{displaymath}
     -->
    </p>
    <div class="mathdisplay unidade" id="2P215" title="2P215">
    <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">&nbsp; &nbsp;e<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">
    </div>
    
    <p class=" unidade" id="2P216" title="2P216">
    Valem as seguintes relações inversas,
    </p>
    <div class="mathdisplay unidade" id="2P217" title="2P217"><table class="equation">
    <tbody><tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img991.svg" alt="$\displaystyle {\mathrm{senh}}({\mathrm{senh}}^{-1} u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img992.svg" alt="$\displaystyle \quad u \in \mathbb{R},$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img993.svg" alt="$\displaystyle {\mathrm{senh}}^{-1}({\mathrm{senh}}u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img994.svg" alt="$\displaystyle \quad u \in \mathbb{R}.$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P218" title="2P218">
    O gráfico da função seno hiperbólico inverso, é da forma,
    </p>
    <div class="CENTER"><a id="2735"></a>
    <table id="2I22" title="2I22">
    <caption class="BOTTOM"><strong>Figura 2.22:</strong>
    Gráfico de seno hiperbólico inverso.</caption>
    <tbody><tr><td>
    <div class="CENTER">
    <img src="img/farcsinh.png" alt="Image farcsinh" loading="lazy"> </div></td></tr>
    </tbody></table>
    </div>
    
    <p class=" unidade" id="2P219" title="2P219">
    A função cosseno hiperbólico não é uma função bijetora. Lembremos que seu domínio é <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>, mas sua imagem é o
    subconjunto <!-- MATH
     $[1,\infty) \subset \mathbb{R}$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img995.svg" alt="$[1,\infty) \subset \mathbb{R}$" loading="lazy"></span>. Restringindo o contradomínio a <!-- MATH
     $[1,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img547.svg" alt="$[1,\infty)$" loading="lazy"></span> tornamos esta função sobrejetora.
    Também a função cosseno hiperbólico, definida em todo o domínio <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>, não é injetora. Vamos então restringir o domínio
    desta função ao conjunto dos reais não negativos. Temos assim que a função cosseno hiperbólico é bijetora de
    <!-- MATH
     $[0,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img996.svg" alt="$[0,\infty)$" loading="lazy"></span> em <!-- MATH
     $[1,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img547.svg" alt="$[1,\infty)$" loading="lazy"></span>. Por restrição, podemos então definir a função cosseno hiperbólico inverso, denotada por
    <a name="2737"></a>
    </p>
    <div class="mathdisplay unidade" id="2P220" title="2P220">
    <!-- MATH
     \begin{eqnarray*}
    f : [1,\infty) & \to & [0,\infty) \\
    u & \mapsto & w = f(u) = \cosh^{-1} u,
    \end{eqnarray*}
     -->
    <table cellpadding="0" align="CENTER" width="100%">
    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img997.svg" alt="$\displaystyle f : [1,\infty)$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img998.svg" alt="$\displaystyle [0,\infty)$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img999.svg" alt="$\displaystyle w = f(u) = \cosh^{-1} u,$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    </tbody></table></div>
    <br clear="ALL">
    <p class=" unidade" id="2P221" style="text-indent: 0 !important;" title="2P221">
    e que satisfaz a relação <!-- MATH
     $u = \cosh w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1000.svg" alt="$u = \cosh w$" loading="lazy"></span>. Levando <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> ao infinito, a relação <!-- MATH
     $u = \cosh w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1000.svg" alt="$u = \cosh w$" loading="lazy"></span> nos mostra que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> também vai
    para o infinito. No outro extremo do intervalo de definição, isto é, quando <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> tende para 1 (somente pela direita), a
    mesma relação mostra que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> vai para 0. Então,
    <!-- MATH
     \begin{displaymath}
    \lim_{u \to \infty} \cosh^{-1} u = \infty \qquad \text{e} \qquad \lim_{u \to 1^{+}} \cosh^{-1} u = 0.
    \end{displaymath}
     -->
    </p>
    <div class="mathdisplay unidade" id="2P222" title="2P222">
    <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">&nbsp; &nbsp;e<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">
    </div>
    
    <p class=" unidade" id="2P223" title="2P223">
    O gráfico desta função é a curva da figura abaixo.
    </p>
    <div class="CENTER"><a id="2748"></a>
    <table id="2I23" title="2I23">
    <caption class="BOTTOM"><strong>Figura 2.23:</strong>
    Gráfico da função cosseno hiperbólico inverso.</caption>
    <tbody><tr><td>
    <div class="CENTER">
    <img src="img/farccosh.png" alt="Image farccosh" loading="lazy"> </div></td></tr>
    </tbody></table>
    </div>
    
    <p class=" unidade" id="2P224" title="2P224">
    Ocorrem as seguintes relações inversas,
    </p>
    <div class="mathdisplay unidade" id="2P225" title="2P225"><table class="equation">
    <tbody><tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1003.svg" alt="$\displaystyle \cosh(\cosh^{-1} u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1004.svg" alt="$\displaystyle \quad u \in [1,\infty),$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1005.svg" alt="$\displaystyle \cosh^{-1}(\cosh u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1006.svg" alt="$\displaystyle \quad u \in [0,\infty).$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P226" title="2P226">
    A função tangente hiperbólica é uma função injetora do conjunto <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span> no conjunto <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>, mas não é sobrejetora já que o
    conjunto imagem é o intervalo <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span>. Restringindo o contradomínio temos a bijetividade da função tangente
    hiperbólica de <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span> em <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span>. Definimos então a função tangente hiperbólica inversa, <a name="2756"></a>
    </p>
    <div class="mathdisplay unidade" id="2P227" title="2P227">
    <!-- MATH
     \begin{eqnarray*}
    f : (-1,1) & \to & \mathbb{R}\\
    u & \mapsto & w = f(u) = {\mathrm {tgh}}^{-1} u,
    \end{eqnarray*}
     -->
    <table cellpadding="0" align="CENTER" width="100%">
    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1007.svg" alt="$\displaystyle f : (-1,1)$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img212.svg" alt="$\displaystyle \mathbb{R}$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1008.svg" alt="$\displaystyle w = f(u) = {\mathrm {tgh}}^{-1} u,$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    </tbody></table></div>
    <br clear="ALL">
    <p class=" unidade" id="2P228" style="text-indent: 0 !important;" title="2P228">
    com <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> satisfazendo <!-- MATH
     $u = {\mathrm {tgh}}w$
     -->
    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1009.svg" alt="$u = {\mathrm {tgh}}w$" loading="lazy"></span>. Vamos observar o seu comportamento nos extremos do intervalo. Quando <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> tende
    a 1 (pela esquerda) então a relação <!-- MATH
     $u = {\mathrm {tgh}}w$
     -->
    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1009.svg" alt="$u = {\mathrm {tgh}}w$" loading="lazy"></span> mostra que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> deve ir para o infinito. Analogamente se <span class="MATH"><img style="height: 1.83ex; vertical-align: -0.27ex; " src="img/img1010.svg" alt="$u \to -1$" loading="lazy"></span>
    então <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> vai para <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img1011.svg" alt="$-\infty$" loading="lazy"></span>. Resumindo,
    <!-- MATH
     \begin{displaymath}
    \lim_{u \to 1^{-}} {\mathrm {tgh}}^{-1} u = \infty \qquad \text{e} \qquad \lim_{u \to -1^{+}} {\mathrm {tgh}}^{-1} u = -\infty.
    \end{displaymath}
     -->
    </p>
    <div class="mathdisplay unidade" id="2P229" title="2P229">
    <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">&nbsp; &nbsp;e<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">
    </div>
    
    <p class=" unidade" id="2P230" title="2P230">
    O gráfico da função tangente hiperbólica inversa,
    </p>
    <div class="CENTER"><a id="2767"></a>
    <table id="2I24" title="2I24">
    <caption class="BOTTOM"><strong>Figura 2.24:</strong>
    Gráfico da função tangente hiperbólica inversa. </caption>
    <tbody><tr><td>
    <div class="CENTER">
    <img src="img/farctgh.png" alt="Image farctgh" loading="lazy"> </div></td></tr>
    </tbody></table>
    </div>
    
    <p class=" unidade" id="2P231" title="2P231">
    As relações inversas são
    </p>
    <div class="mathdisplay unidade" id="2P232" title="2P232"><table class="equation">
    <tbody><tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1014.svg" alt="$\displaystyle {\mathrm {tgh}}({\mathrm {tgh}}^{-1} u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1015.svg" alt="$\displaystyle \quad u \in (-1,1),$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1016.svg" alt="$\displaystyle {\mathrm {tgh}}^{-1}({\mathrm {tgh}}u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img994.svg" alt="$\displaystyle \quad u \in \mathbb{R}.$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P233" title="2P233">
    A função cotangente hiperbólica também é uma função bijetora do conjunto <!-- MATH
     $\mathbb{R}^{*}$
     -->
    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span> no conjunto <!-- MATH
     $(-\infty,-1) \cup
    (1,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img619.svg" alt="$(-\infty,-1) \cup (1,\infty)$" loading="lazy"></span>. Desta forma, definimos a função cotangente hiperbólica inversa por, <a name="2776"></a>
    </p>
    <div class="mathdisplay unidade" id="2P234" title="2P234">
    <!-- MATH
     \begin{eqnarray*}
    f : (-\infty,-1) \cup (1,\infty) & \to & \mathbb{R}-\{0\} \\
    u & \mapsto & w = f(u) = {\mathrm{ctgh}}^{-1} u,
    \end{eqnarray*}
     -->
    <table cellpadding="0" align="CENTER" width="100%">
    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1017.svg" alt="$\displaystyle f : (-\infty,-1) \cup (1,\infty)$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1018.svg" alt="$\displaystyle \mathbb{R}-\{0\}$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1019.svg" alt="$\displaystyle w = f(u) = {\mathrm{ctgh}}^{-1} u,$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    </tbody></table></div>
    <br clear="ALL">
    <p class=" unidade" id="2P235" style="text-indent: 0 !important;" title="2P235">
    desde que <!-- MATH
     $u = {\mathrm{ctgh}}w$
     -->
    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1020.svg" alt="$u = {\mathrm{ctgh}}w$" loading="lazy"></span>. Analisando os extremos do intervalo de definição, temos que quando <!-- MATH
     $u \to -\infty$
     -->
    <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img865.svg" alt="$u \to -\infty$" loading="lazy"></span> a relação
    <!-- MATH
     $u = {\mathrm{ctgh}}w$
     -->
    <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1020.svg" alt="$u = {\mathrm{ctgh}}w$" loading="lazy"></span> nos diz que isto ocorre quando <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> vai para 0 (com valores negativos). Analogamente, quando <!-- MATH
     $u \to
    \infty$
     -->
    <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img863.svg" alt="$u \to \infty$" loading="lazy"></span> então deve ocorrer <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1021.svg" alt="$w \to 0$" loading="lazy"></span> (com valores positivos). Fazendo <!-- MATH
     $u \to -1^{-}$
     -->
    <span class="MATH"><img style="height: 2.03ex; vertical-align: -0.27ex; " src="img/img1022.svg" alt="$u \to -1^{-}$" loading="lazy"></span> então, a mesma relação anterior,
    nos diz que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> deve ir para <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img1011.svg" alt="$-\infty$" loading="lazy"></span> e analogamente <!-- MATH
     $w \to \infty$
     -->
    <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1023.svg" alt="$w \to \infty$" loading="lazy"></span> quando <!-- MATH
     $u \to 1^{+}$
     -->
    <span class="MATH"><img style="height: 1.86ex; vertical-align: -0.11ex; " src="img/img1024.svg" alt="$u \to 1^{+}$" loading="lazy"></span>. Resumindo,
    </p>
    <div class="mathdisplay unidade" id="2P236" title="2P236"><table class="equation">
    <tbody><tr>
    <td style="text-align:center;"><span class="MATH"><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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td style="text-align:center;"><span class="MATH"><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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td style="text-align:center;"><span class="MATH"><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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P237" title="2P237">
    O gráfico desta função é dado por
    </p>
    <div class="CENTER"><a id="2794"></a>
    <table id="2I25" title="2I25">
    <caption class="BOTTOM"><strong>Figura 2.25:</strong>
    Gráfico da função cotangente hiperbólica inversa.</caption>
    <tbody><tr><td>
    <div class="CENTER">
    <img src="img/farcctgh.png" alt="Image farcctgh" loading="lazy"> </div></td></tr>
    </tbody></table>
    </div>
    
    <p class=" unidade" id="2P238" title="2P238">
    Valem as seguintes relações de inversão,
    </p>
    <div class="mathdisplay unidade" id="2P239" title="2P239"><table class="equation">
    <tbody><tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1028.svg" alt="$\displaystyle {\mathrm{ctgh}}({\mathrm{ctgh}}^{-1} u) = u$" loading="lazy">&nbsp; &nbsp;para todo<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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1030.svg" alt="$\displaystyle {\mathrm{ctgh}}^{-1}({\mathrm{ctgh}}u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1031.svg" alt="$\displaystyle \quad u \in \mathbb{R}-\{0\}.$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P240" title="2P240">
    Para a secante hiperbólica, temos alguns problemas como no caso do cosseno hiperbólico inverso. O domínio da
    função secante hiperbólica é o conjunto <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span> e a imagem é o conjunto <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img887.svg" alt="$(0,1]$" loading="lazy"></span>. Mas esta função não é injetora de <!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span>
    em <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img887.svg" alt="$(0,1]$" loading="lazy"></span>. Então vamos restringir o conjunto domínio para os reais não negativos. Assim, a função secante hiperbólica
    é bijetiva de <!-- MATH
     $[0,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img996.svg" alt="$[0,\infty)$" loading="lazy"></span> em <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img887.svg" alt="$(0,1]$" loading="lazy"></span> e podemos definir a função secante hiperbólica inversa <a name="2802"></a>
    </p>
    <div class="mathdisplay unidade" id="2P241" title="2P241">
    <!-- MATH
     \begin{eqnarray*}
    f: (0,1] & \to & [0,\infty) \\
    u & \mapsto & w = f(u) = {\mathrm{sech}}^{-1} u,
    \end{eqnarray*}
     -->
    <table cellpadding="0" align="CENTER" width="100%">
    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1032.svg" alt="$\displaystyle f: (0,1]$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img998.svg" alt="$\displaystyle [0,\infty)$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1033.svg" alt="$\displaystyle w = f(u) = {\mathrm{sech}}^{-1} u,$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    </tbody></table></div>
    <br clear="ALL">
    <p class=" unidade" id="2P242" style="text-indent: 0 !important;" title="2P242">
    satisfazendo <!-- MATH
     $u = {\mathrm{sech}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1034.svg" alt="$u = {\mathrm{sech}}w$" loading="lazy"></span>. Quando <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img270.svg" alt="$u \to 0$" loading="lazy"></span> (pela direita), a relação <!-- MATH
     $u = {\mathrm{sech}}w = \frac{1}{\cosh w}$
     -->
    <span class="MATH"><img style="height: 2.83ex; vertical-align: -0.85ex; " src="img/img1035.svg" alt="$u = {\mathrm{sech}}w = \frac{1}{\cosh w}$" loading="lazy"></span> diz que <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1036.svg" alt="$\cosh
    w$" loading="lazy"></span> deve estar indo para o infinito por valores positivos e consequentemente <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> deve estar indo para o infinito.
    Quando <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> vai para 1 (pela esquerda) então <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1036.svg" alt="$\cosh
    w$" loading="lazy"></span> está indo para 1 e <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> deve estar se aproximando de 0. Temos
    então
    <!-- MATH
     \begin{displaymath}
    \lim_{u \to 0^{+}} {\mathrm{sech}}^{-1} u = \infty \qquad \text{e} \qquad \lim_{u \to 1^{-}} {\mathrm{sech}}^{-1} u = 0.
    \end{displaymath}
     -->
    </p>
    <div class="mathdisplay unidade" id="2P243" title="2P243">
    <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">&nbsp; &nbsp;e<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">
    </div>
    
    <p class=" unidade" id="2P244" title="2P244">
    As relações inversas ficam,
    </p>
    <div class="mathdisplay unidade" id="2P245" title="2P245"><table class="equation">
    <tbody><tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1039.svg" alt="$\displaystyle {\mathrm{sech}}({\mathrm{sech}}^{-1} u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1040.svg" alt="$\displaystyle \quad u \in (0,1],$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1041.svg" alt="$\displaystyle {\mathrm{sech}}^{-1}({\mathrm{sech}}u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1006.svg" alt="$\displaystyle \quad u \in [0,\infty).$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P246" title="2P246">
    Graficamente, temos
    </p>
    <div class="CENTER"><a id="2821"></a>
    <table id="2I26" title="2I26">
    <caption class="BOTTOM"><strong>Figura 2.26:</strong>
    Gráfico de secante hiperbólica inversa.</caption>
    <tbody><tr><td>
    <div class="CENTER">
    <img src="img/farcsech.png" alt="Image farcsech" loading="lazy"> </div></td></tr>
    </tbody></table>
    </div>
    
    <p class=" unidade" id="2P247" title="2P247">
    Finalmente, lembremos que a função cossecante hiperbólica é bijetora do conjunto <!-- MATH
     $\mathbb{R}^{*}$
     -->
    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span> no conjunto <!-- MATH
     $\mathbb{R}^{*}$
     -->
    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span>.
    Definimos então a função cossecante hiperbólica inversa <a name="2825"></a>
    </p>
    <div class="mathdisplay unidade" id="2P248" title="2P248">
    <!-- MATH
     \begin{eqnarray*}
    f: \mathbb{R}-\{0\} & \to & \mathbb{R}-\{0\} \\
    u & \mapsto & w = f(u) = {\mathrm{csch}}^{-1} u,
    \end{eqnarray*}
     -->
    <table cellpadding="0" align="CENTER" width="100%">
    <tbody><tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1042.svg" alt="$\displaystyle f: \mathbb{R}-\{0\}$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img211.svg" alt="$\displaystyle \to$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1018.svg" alt="$\displaystyle \mathbb{R}-\{0\}$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    <tr valign="MIDDLE"><td nowrap="" width="50%" align="RIGHT"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img213.svg" alt="$\displaystyle u$" loading="lazy"></td>
    <td width="10" align="CENTER" nowrap=""><img style="height: 0.97ex; vertical-align: -0.10ex; " src="img/img214.svg" alt="$\displaystyle \mapsto$" loading="lazy"></td>
    <td align="LEFT" nowrap="" width="50%"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1043.svg" alt="$\displaystyle w = f(u) = {\mathrm{csch}}^{-1} u,$" loading="lazy"></td>
    <td class="eqno" width="10" align="RIGHT">
    &nbsp;</td></tr>
    </tbody></table></div>
    <br clear="ALL">
    <p class=" unidade" id="2P249" style="text-indent: 0 !important;" title="2P249">
    que também satisfaz <!-- MATH
     $u = {\mathrm{csch}}w = \frac{1}{{\mathrm{senh}}w}$
     -->
    <span class="MATH"><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"></span>. Esta relação explica também os limites. Quando <!-- MATH
     $u \to -\infty$
     -->
    <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img865.svg" alt="$u \to -\infty$" loading="lazy"></span>
    então <!-- MATH
     ${\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1045.svg" alt="${\mathrm{senh}}w$" loading="lazy"></span> deve ir para 0 por valores negativos e então <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> deve ir para 0 também por valores negativos.
    Analogamente, quando <!-- MATH
     $u \to \infty$
     -->
    <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img863.svg" alt="$u \to \infty$" loading="lazy"></span>, <!-- MATH
     ${\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1045.svg" alt="${\mathrm{senh}}w$" loading="lazy"></span> deve ir para 0 por valores positivos e então <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> deve ir também para 0
    por valores positivos. Se <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img270.svg" alt="$u \to 0$" loading="lazy"></span> por valores positivos então <!-- MATH
     ${\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1045.svg" alt="${\mathrm{senh}}w$" loading="lazy"></span> deve ir para o infinito e <!-- MATH
     $w \to \infty$
     -->
    <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1023.svg" alt="$w \to \infty$" loading="lazy"></span>
    também. Da mesma forma, se <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img270.svg" alt="$u \to 0$" loading="lazy"></span> por valores negativos, então <!-- MATH
     ${\mathrm{senh}}w$
     -->
    <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1045.svg" alt="${\mathrm{senh}}w$" loading="lazy"></span> vai para <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img1011.svg" alt="$-\infty$" loading="lazy"></span> e consequentemente, <!-- MATH
     $w
    \to -\infty$
     -->
    <span class="MATH"><img style="height: 1.59ex; vertical-align: -0.27ex; " src="img/img1046.svg" alt="$w
    \to -\infty$" loading="lazy"></span> também. Resumindo,
    </p>
    <div class="mathdisplay unidade" id="2P250" title="2P250"><table class="equation">
    <tbody><tr>
    <td style="text-align:center;"><span class="MATH"><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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td style="text-align:center;"><span class="MATH"><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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td style="text-align:center;"><span class="MATH"><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"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P251" title="2P251">
    O gráfico desta função é representado por
    </p>
    <div class="CENTER"><a id="2843"></a>
    <table id="2I27" title="2I27">
    <caption class="BOTTOM"><strong>Figura 2.27:</strong>
    Gráfico da função cossecante hiperbólica inversa.</caption>
    <tbody><tr><td>
    <div class="CENTER">
    <img src="img/farccsch.png" alt="Image farccsch" loading="lazy"> </div></td></tr>
    </tbody></table>
    </div>
    
    <p class=" unidade" id="2P252" title="2P252">
    São válidas as relações de inversão,
    </p>
    <div class="mathdisplay unidade" id="2P253" title="2P253"><table class="equation">
    <tbody><tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1050.svg" alt="$\displaystyle {\mathrm{csch}}({\mathrm{csch}}^{-1} u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 1.86ex; vertical-align: -0.18ex; " src="img/img1051.svg" alt="$\displaystyle \quad u \in \mathbb{R}^{*},$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    <tr>
    <td>&nbsp;</td>
    <td style="text-align:left;"><span class="MATH"><img style="height: 2.65ex; vertical-align: -0.62ex; " src="img/img1052.svg" alt="$\displaystyle {\mathrm{csch}}^{-1}({\mathrm{csch}}u) = u$" loading="lazy">&nbsp; &nbsp;para todo<img style="height: 1.86ex; vertical-align: -0.18ex; " src="img/img1053.svg" alt="$\displaystyle \quad u \in \mathbb{R}^{*}.$" loading="lazy"></span></td>
    <td class="eqno" style="text-align:right">
    &nbsp;&nbsp;&nbsp;</td></tr>
    </tbody></table></div>
    
    
    <p class=" unidade" id="2P254" title="2P254">
    A relação completa de funções trigonométricas hiperbólicas inversas com seus respectivos domínios de definição e
    conjunto imagem é dada na próxima tabela.
    
    </p>
    <div class="CENTER"><a id="2868"></a>
    <table id="2T3" title="2T3">
    <caption><strong>Tabela 2.3:</strong>
    Domínio e imagem das funções trigonométricas hiperbólicas inversas.</caption>
    <tbody><tr><td>
    <div class="CENTER">
    <table class="PAD  BORDER">
    <tbody><tr><td class="LEFT">função</td>
    <td class="CENTER">domínio</td>
    <td class="CENTER">imagem</td>
    </tr>
    <tr><td class="LEFT"><!-- MATH
     ${\mathrm{senh}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img986.svg" alt="${\mathrm{senh}}^{-1} u$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
    </tr>
    <tr><td class="LEFT"><!-- MATH
     $\cosh^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img1054.svg" alt="$\cosh^{-1} u$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $[1,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img547.svg" alt="$[1,\infty)$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $[0,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img996.svg" alt="$[0,\infty)$" loading="lazy"></span></td>
    </tr>
    <tr><td class="LEFT"><!-- MATH
     ${\mathrm {tgh}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.59ex; vertical-align: -0.58ex; " src="img/img1055.svg" alt="${\mathrm {tgh}}^{-1} u$" loading="lazy"></span></td>
    <td class="CENTER"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $\mathbb{R}$
     -->
    <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
    </tr>
    <tr><td class="LEFT"><!-- MATH
     ${\mathrm{ctgh}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.59ex; vertical-align: -0.58ex; " src="img/img1056.svg" alt="${\mathrm{ctgh}}^{-1} u$" loading="lazy"></span></td>
    <td class="CENTER">&nbsp;&nbsp;&nbsp;&nbsp;<!-- MATH
     $(-\infty,-1) \cup (1,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img619.svg" alt="$(-\infty,-1) \cup (1,\infty)$" loading="lazy"></span> &nbsp;&nbsp;&nbsp;&nbsp;</td>
    <td class="CENTER"><!-- MATH
     $\mathbb{R}^{*}$
     -->
    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span></td>
    </tr>
    <tr><td class="LEFT"><!-- MATH
     ${\mathrm{sech}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img1057.svg" alt="${\mathrm{sech}}^{-1} u$" loading="lazy"></span></td>
    <td class="CENTER"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img887.svg" alt="$(0,1]$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $[0,\infty)$
     -->
    <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img996.svg" alt="$[0,\infty)$" loading="lazy"></span></td>
    </tr>
    <tr><td class="LEFT"><!-- MATH
     ${\mathrm{csch}}^{-1} u$
     -->
    <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img1058.svg" alt="${\mathrm{csch}}^{-1} u$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $\mathbb{R}^{*}$
     -->
    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span></td>
    <td class="CENTER"><!-- MATH
     $\mathbb{R}^{*}$
     -->
    <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span></td>
    </tr>
    </tbody></table>
    </div></td></tr>
    </tbody></table>
    </div>

:::

## 2.7 Continuidade das funções trigonométricas hiperbólicas inversas {#SECTION00670000000000000000}

::: {.raw_html}
        
        <p class=" unidade" id="2P255" title="2P255">
        O procedimento adotado aqui não tem diferenças do procedimento adotado para as funções trigonométricas circulares. O
        teorema <a href="/trigonometria-hiperbolica/funcoes-trigonometricas-circulares#teoinvcont">1.11</a> se aplica às funções trigonométricas hiperbólicas em seus respectivos domínios de definição.
        Vamos omitir os detalhes. Entretanto entendemos deste ponto em diante que cada função trigonométrica inversa é contínua
        nos seus respectivos domínios de definição respeitando a lateralidade nos extremos fechados destes domínios.
        </p>
        <p class=" unidade" id="2P256" title="2P256">
        Temos assim, que
        </p>
        <div class="CENTER">
        <table class="PAD ">
        <tbody><tr><td class="LEFT"><!-- MATH
         $\lim\limits_{u \to a} {\mathrm{senh}}^{-1} u = {\mathrm{senh}}^{-1} a$
         -->
        <span class="MATH"><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"></span></td>
        <td class="CENTER">para todo</td>
        <td class="LEFT"><!-- MATH
         $a \in \mathbb{R}$
         -->
        <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img284.svg" alt="$a \in \mathbb{R}$" loading="lazy"></span>,</td>
        </tr>
        <tr><td class="LEFT"><!-- MATH
         $\lim\limits_{u \to a} \cosh^{-1} u = \cosh^{-1} a$
         -->
        <span class="MATH"><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"></span></td>
        <td class="CENTER">para todo</td>
        <td class="LEFT"><!-- MATH
         $a \in [1,\infty)$
         -->
        <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1061.svg" alt="$a \in [1,\infty)$" loading="lazy"></span>,</td>
        </tr>
        <tr><td class="LEFT"><!-- MATH
         $\lim\limits_{u \to a} {\mathrm {tgh}}^{-1} u = {\mathrm {tgh}}^{-1} a$
         -->
        <span class="MATH"><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"></span></td>
        <td class="CENTER">para todo</td>
        <td class="LEFT"><!-- MATH
         $a \in (-1,1)$
         -->
        <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1063.svg" alt="$a \in (-1,1)$" loading="lazy"></span>,</td>
        </tr>
        <tr><td class="LEFT"><!-- MATH
         $\lim\limits_{u \to a} {\mathrm{ctgh}}^{-1} u = {\mathrm{ctgh}}^{-1} a$
         -->
        <span class="MATH"><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"></span></td>
        <td class="CENTER">para todo</td>
        <td class="LEFT"><!-- MATH
         $a \in (-\infty,-1) \cup (1,\infty)$
         -->
        <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1065.svg" alt="$a \in (-\infty,-1) \cup (1,\infty)$" loading="lazy"></span>,</td>
        </tr>
        <tr><td class="LEFT"><!-- MATH
         $\lim\limits_{u \to a} {\mathrm{sech}}^{-1} u = {\mathrm{sech}}^{-1} a$
         -->
        <span class="MATH"><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"></span></td>
        <td class="CENTER">para todo</td>
        <td class="LEFT"><!-- MATH
         $a \in (0,1]$
         -->
        <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1067.svg" alt="$a \in (0,1]$" loading="lazy"></span>,</td>
        </tr>
        <tr><td class="LEFT"><!-- MATH
         $\lim\limits_{u \to a} {\mathrm{csch}}^{-1} u = {\mathrm{csch}}^{-1} u$
         -->
        <span class="MATH"><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"></span></td>
        <td class="CENTER">para todo</td>
        <td class="LEFT"><!-- MATH
         $a \in \mathbb{R}^{*}$
         -->
        <span class="MATH"><img style="height: 1.75ex; vertical-align: -0.18ex; " src="img/img1069.svg" alt="$a \in \mathbb{R}^{*}$" loading="lazy"></span>.</td>
        </tr>
        </tbody></table>
        </div>
               
:::

## 2.8 Derivadas das funções trigonométricas hiperbólicas inversas {#SECTION00680000000000000000}

::: {.raw_html}

       <a id="secderhipinv">
            
            <p class=" unidade" id="2P257" title="2P257">
            Nesta seção, vamos determinar as fórmulas de derivada para as funções trigonométricas hiperbólicas inversas. Usaremos
            principalmente a técnica da diferenciação implícita e levamos em conta o conhecimento das fórmulas de diferenciação
            para as seis funções trigonométricas hiperbólicas obtidas na seção <a href="#secderhip">2.5</a>.
            </p>
            <p class=" unidade" id="2P258" title="2P258">
            Considerando a função <!-- MATH
             $w = f(u) = {\mathrm{senh}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.63ex; vertical-align: -0.62ex; " src="img/img1070.svg" alt="$w = f(u) = {\mathrm{senh}}^{-1} u$" loading="lazy"></span>, para todo <!-- MATH
             $u \in \mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>, queremos agora derivar em relação a <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e obter
            <!-- MATH
             $w' = \frac{dw}{du}$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img556.svg" alt="$w' = \frac{dw}{du}$" loading="lazy"></span>. Sabemos que neste caso é válida a relação <!-- MATH
             $u = {\mathrm{senh}}w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img984.svg" alt="$u = {\mathrm{senh}}w$" loading="lazy"></span>. Lembre-se que <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> é variável dependente
            de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> e, por isto, quando derivarmos <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> devemos usar diferenciação implícita. Nestes termos, derivando em relação a
            <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> os dois membros de <!-- MATH
             $u = {\mathrm{senh}}w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img984.svg" alt="$u = {\mathrm{senh}}w$" loading="lazy"></span>, temos
            <!-- MATH
             \begin{displaymath}
            1 = \frac{d}{du} {\mathrm{senh}}w = \cosh w \cdot \frac{dw}{du}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P259" title="2P259">
            <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">
            </div>
            
            <p class=" unidade" id="2P260" title="2P260">
            Como queremos determinar <!-- MATH
             $w' = \frac{dw}{du}$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img556.svg" alt="$w' = \frac{dw}{du}$" loading="lazy"></span> basta agora isolar este termo. Obtemos
            <!-- MATH
             \begin{displaymath}
            \frac{dw}{du} = \frac{1}{\cosh w}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P261" title="2P261">
            <img style="height: 4.55ex; vertical-align: -1.57ex; " src="img/img1072.svg" alt="$\displaystyle \frac{dw}{du} = \frac{1}{\cosh w}. $" loading="lazy">
            </div>
            
            <p class=" unidade" id="2P262" title="2P262">
            Mas claro que desejamos obter esta derivada como função de <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span> novamente. Precisamos então substituir a variável
            dependente <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img434.svg" alt="$w$" loading="lazy"></span> do segundo membro pela variável independente <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>. A única expressão que faz esta substituição é a
            própria relação <!-- MATH
             $u = {\mathrm{senh}}w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img984.svg" alt="$u = {\mathrm{senh}}w$" loading="lazy"></span>. Assim, vamos substituir o termo <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1036.svg" alt="$\cosh
            w$" loading="lazy"></span> por alguma expressão que contenha <!-- MATH
             ${\mathrm{senh}}w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1045.svg" alt="${\mathrm{senh}}w$" loading="lazy"></span>.
            Usando a relação fundamental (<a href="#idfundhip">2.3</a>), temos
            <!-- MATH
             \begin{displaymath}
            \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}}},
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P263" title="2P263">
            <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...
            ...w} = \frac{1}{\sqrt{ 1 + {\mathrm{senh}}^{2} w}} = \frac{1}{\sqrt{1 + u^{2}}}, $" loading="lazy">
            </div><p class=" unidade" id="2P264" style="text-indent: 0 !important;" title="2P264">
            para todo <!-- MATH
             $u \in \mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.73ex; vertical-align: -0.18ex; " src="img/img57.svg" alt="$u \in \mathbb{R}$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P265" title="2P265">
            Tomamos agora a função <!-- MATH
             $w = f(u) = \cosh^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.63ex; vertical-align: -0.62ex; " src="img/img1074.svg" alt="$w = f(u) = \cosh^{-1} u$" loading="lazy"></span>, definida para todo <!-- MATH
             $u \in [1,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1075.svg" alt="$u \in [1,\infty)$" loading="lazy"></span>. Derivando implicitamente a
            igualdade <!-- MATH
             $u = \cosh w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1000.svg" alt="$u = \cosh w$" loading="lazy"></span> com relação a <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, para todo <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img1076.svg" alt="$u > 1$" loading="lazy"></span>, obtemos
            <!-- MATH
             \begin{displaymath}
            1 = ({\mathrm{senh}}w) \cdot \frac{dw}{du}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P266" title="2P266">
            <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">
            </div>
            
            <p class=" unidade" id="2P267" title="2P267">
            Isolando agora o termo <!-- MATH
             $\frac{dw}{du}$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img1078.svg" alt="$\frac{dw}{du}$" loading="lazy"></span>, como feito para o caso do seno hiperbólico e usando a relação fundamental
            (<a href="#idfundhip">2.3</a>), obtemos
            <!-- MATH
             \begin{displaymath}
            \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}}
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P268" title="2P268">
            <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}} $" loading="lazy">
            </div><p class=" unidade" id="2P269" style="text-indent: 0 !important;" title="2P269">
            para todo <span class="MATH"><img style="height: 1.69ex; vertical-align: -0.14ex; " src="img/img1076.svg" alt="$u > 1$" loading="lazy"></span>. Note que esta derivada não está definida para <span class="MATH"><img style="height: 1.66ex; vertical-align: -0.11ex; " src="img/img309.svg" alt="$u=1$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P270" title="2P270">
            Para a função <!-- MATH
             $w = f(u) = {\mathrm {tgh}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.63ex; vertical-align: -0.62ex; " src="img/img1080.svg" alt="$w = f(u) = {\mathrm {tgh}}^{-1} u$" loading="lazy"></span>, definida no intervalo <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span>, derivamos a igualdade <!-- MATH
             $u = {\mathrm {tgh}}w$
             -->
            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1009.svg" alt="$u = {\mathrm {tgh}}w$" loading="lazy"></span> com relação
            a <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, obtendo
            <!-- MATH
             \begin{displaymath}
            1 = ({\mathrm{sech}}^{2} w) \cdot \frac{dw}{du}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P271" title="2P271">
            <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">
            </div>
            
            <p class=" unidade" id="2P272" title="2P272">
            Reorganizando os termos e usando a igualdade (<a href="#idtghsech">2.8</a>), da proposição <a href="#propidsectghip">2.3</a>, vem
            <!-- MATH
             \begin{displaymath}
            \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}},
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P273" title="2P273">
            <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">
            </div><p class=" unidade" id="2P274" style="text-indent: 0 !important;" title="2P274">
            para todo <!-- MATH
             $u \in (-1,1)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img552.svg" alt="$u \in (-1,1)$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P275" title="2P275">
            Considerando <!-- MATH
             $w = f(u) = {\mathrm{ctgh}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.63ex; vertical-align: -0.62ex; " src="img/img1083.svg" alt="$w = f(u) = {\mathrm{ctgh}}^{-1} u$" loading="lazy"></span>, definida para todo <!-- MATH
             $u \in (-\infty,-1) \cup (1,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img583.svg" alt="$u \in
            (-\infty,-1) \cup (1,\infty)$" loading="lazy"></span>, vamos derivar a
            igualdade <!-- MATH
             $u = {\mathrm{ctgh}}w$
             -->
            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1020.svg" alt="$u = {\mathrm{ctgh}}w$" loading="lazy"></span> com respeito a <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>. Obtemos
            <!-- MATH
             \begin{displaymath}
            1 = (-{\mathrm{csch}}^{2} w) \cdot \frac{dw}{du}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P276" title="2P276">
            <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">
            </div>
            
            <p class=" unidade" id="2P277" title="2P277">
            Isolando o termo <!-- MATH
             $w' = \frac{dw}{du}$
             -->
            <span class="MATH"><img style="height: 2.81ex; vertical-align: -0.83ex; " src="img/img556.svg" alt="$w' = \frac{dw}{du}$" loading="lazy"></span> e usando a identidade (<a href="#idctghcsch">2.9</a>) da proposição <a href="#propidsectghip">2.3</a>,
            temos
            <!-- MATH
             \begin{displaymath}
            \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}},
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P278" title="2P278">
            <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}{{...
            ...frac{-1}{{\mathrm{ctgh}}^{2}w - 1} = \frac{-1}{u^{2}-1} = \frac{1}{1 - u^{2}}, $" loading="lazy">
            </div><p class=" unidade" id="2P279" style="text-indent: 0 !important;" title="2P279">
            para <!-- MATH
             $u \in (-\infty,-1) \cup (1,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img583.svg" alt="$u \in
            (-\infty,-1) \cup (1,\infty)$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P280" title="2P280">
            Tomando agora a função <!-- MATH
             $w = f(u) = {\mathrm{sech}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.63ex; vertical-align: -0.62ex; " src="img/img1086.svg" alt="$w = f(u) = {\mathrm{sech}}^{-1} u$" loading="lazy"></span>, que está definida para todo <!-- MATH
             $u \in (0,1]$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1087.svg" alt="$u \in (0,1]$" loading="lazy"></span>, temos <!-- MATH
             $u = {\mathrm{sech}}w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1034.svg" alt="$u = {\mathrm{sech}}w$" loading="lazy"></span>, com
            <!-- MATH
             $w \in [0,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1088.svg" alt="$w \in [0,\infty)$" loading="lazy"></span>. Derivando em relação a <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, obtemos
            <!-- MATH
             \begin{displaymath}
            1 = -({\mathrm{sech}}w {\mathrm {tgh}}w) \cdot \frac{dw}{du},
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P281" title="2P281">
            <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">
            </div><p class=" unidade" id="2P282" style="text-indent: 0 !important;" title="2P282">
            para todo <!-- MATH
             $u \in (0,1)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img906.svg" alt="$u \in (0,1)$" loading="lazy"></span>. Então,
            <!-- MATH
             \begin{displaymath}
            \frac{dw}{du} = \frac{-1}{{\mathrm{sech}}w {\mathrm {tgh}}w}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P283" title="2P283">
            <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">
            </div>
            
            <p class=" unidade" id="2P284" title="2P284">
            Usaremos a identidade (<a href="#idtghsech">2.8</a>) da proposição <a href="#propidsectghip">2.3</a>, válida para <!-- MATH
             $w \in [0,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1088.svg" alt="$w \in [0,\infty)$" loading="lazy"></span>. Extraindo a
            raiz quadrada em ambos os membros de (<a href="#idtghsech">2.8</a>), temos que
            <!-- MATH
             \begin{displaymath}
            |{\mathrm {tgh}}w| = \sqrt{1-{\mathrm{sech}}^{2} w}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P285" title="2P285">
            <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">
            </div>
            
            <p class=" unidade" id="2P286" title="2P286">
            Como <!-- MATH
             $w \in (0,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1092.svg" alt="$w \in (0,\infty)$" loading="lazy"></span> o termo <!-- MATH
             ${\mathrm {tgh}}w$
             -->
            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1093.svg" alt="${\mathrm {tgh}}w$" loading="lazy"></span> do lado esquerdo é sempre positivo. Descartamos então o módulo, obtendo
            <!-- MATH
             \begin{displaymath}
            \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}}}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P287" title="2P287">
            <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}{{\...
            ...hrm{sech}}w \sqrt{1 - {\mathrm{sech}}^{2} w}} = \frac{-1}{u \sqrt{1 - u^{2}}}. $" loading="lazy">
            </div>
            
            <p class=" unidade" id="2P288" title="2P288">
            Finalmente para a função <!-- MATH
             $w = f(u) = {\mathrm{csch}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.63ex; vertical-align: -0.62ex; " src="img/img1095.svg" alt="$w = f(u) = {\mathrm{csch}}^{-1} u$" loading="lazy"></span>, definida para todo <!-- MATH
             $u \in \mathbb{R}-\{0\}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img824.svg" alt="$u \in \mathbb{R}-\{0\}$" loading="lazy"></span>, escrevemos <!-- MATH
             $u = {\mathrm{csch}}w$
             -->
            <span class="MATH"><img style="height: 1.68ex; vertical-align: -0.13ex; " src="img/img1096.svg" alt="$u = {\mathrm{csch}}w$" loading="lazy"></span>, com
            <!-- MATH
             $w \in \mathbb{R}-\{0\}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1097.svg" alt="$w \in \mathbb{R}-\{0\}$" loading="lazy"></span> e derivando implicitamente em relação a <span class="MATH"><img style="height: 1.16ex; vertical-align: -0.10ex; " src="img/img1.svg" alt="$u$" loading="lazy"></span>, obtemos
            <!-- MATH
             \begin{displaymath}
            1 = - ({\mathrm{csch}}w {\mathrm{ctgh}}w) \cdot \frac{dw}{du},
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P289" title="2P289">
            <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}, $" loading="lazy">
            </div><p class=" unidade" id="2P290" style="text-indent: 0 !important;" title="2P290">
            que nos fornece
            <!-- MATH
             \begin{displaymath}
            \frac{dw}{du} = \frac{-1}{{\mathrm{csch}}w {\mathrm{ctgh}}w}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P291" title="2P291">
            <img style="height: 5.00ex; vertical-align: -2.03ex; " src="img/img1099.svg" alt="$\displaystyle \frac{dw}{du} = \frac{-1}{{\mathrm{csch}}w {\mathrm{ctgh}}w}. $" loading="lazy">
            </div>
            
            <p class=" unidade" id="2P292" title="2P292">
            Vamos usar a igualdade (<a href="#idctghcsch">2.9</a>), da proposição <a href="#propidsectghip">2.3</a>, válida para <!-- MATH
             $w \in \mathbb{R}^{*}$
             -->
            <span class="MATH"><img style="height: 1.75ex; vertical-align: -0.18ex; " src="img/img1100.svg" alt="$w \in \mathbb{R}^{*}$" loading="lazy"></span>. Extraímos a
            raiz quadrada em ambos os membros de (<a href="#idctghcsch">2.9</a>) para obter
            <!-- MATH
             \begin{displaymath}
            |{\mathrm{ctgh}}w| = \sqrt{1 + {\mathrm{csch}}^{2} w}.
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P293" title="2P293">
            <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">
            </div>
            
            <p class=" unidade" id="2P294" title="2P294">
            Observe que <!-- MATH
             ${\mathrm{ctgh}}w$
             -->
            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1102.svg" alt="${\mathrm{ctgh}}w$" loading="lazy"></span> não é sempre positiva para <!-- MATH
             $w \in \mathbb{R}-\{0\}$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img1097.svg" alt="$w \in \mathbb{R}-\{0\}$" loading="lazy"></span> e isto nos impede de descartar o módulo. Mas <!-- MATH
             ${\mathrm{csch}}
            w {\mathrm{ctgh}}w$
             -->
            <span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1103.svg" alt="${\mathrm{csch}}
            w {\mathrm{ctgh}}w$" loading="lazy"></span> é sempre positivo. Então temos
            </p>
            <div class="mathdisplay unidade" id="2P295" title="2P295"><table class="equation">
            <tbody><tr>
            <td style="text-align:right;"><span class="MATH"><img style="height: 2.13ex; vertical-align: -0.58ex; " src="img/img1104.svg" alt="$\displaystyle {\mathrm{csch}}w {\mathrm{ctgh}}w$" loading="lazy"></span></td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            <tr>
            <td>&nbsp;</td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            <tr>
            <td>&nbsp;</td>
            <td style="text-align:left;"><span class="MATH"><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"></span></td>
            <td class="eqno" style="text-align:right">
            &nbsp;&nbsp;&nbsp;</td></tr>
            </tbody></table></div>
            <p class=" unidade" id="2P296" style="text-indent: 0 !important;" title="2P296">
            donde segue que
            <!-- MATH
             \begin{displaymath}
            \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}}},
            \end{displaymath}
             -->
            </p>
            <div class="mathdisplay unidade" id="2P297" title="2P297">
            <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">
            </div><p class=" unidade" id="2P298" style="text-indent: 0 !important;" title="2P298">
            para todo <!-- MATH
             $u \in \mathbb{R}^{*}$
             -->
            <span class="MATH"><img style="height: 1.75ex; vertical-align: -0.18ex; " src="img/img969.svg" alt="$u \in \mathbb{R}^{*}$" loading="lazy"></span>.
            </p>
            <p class=" unidade" id="2P299" title="2P299">
            Vamos resumir as fórmulas desta seção na próxima tabela.
            </p>
            <div class="CENTER"><a id="3055"></a>
            <table id="2T4" title="2T4">
            <caption><strong>Tabela 2.4:</strong>
            Derivadas das funções trigonométricas hiperbólicas inversas.</caption>
            <tbody><tr><td>
            <div class="CENTER">
            <table class="PAD  BORDER">
            <tbody><tr><td class="LEFT">função</td>
            <td class="CENTER">domínio</td>
            <td class="CENTER">derivada</td>
            </tr>
            <tr><td class="LEFT"><!-- MATH
             ${\mathrm{senh}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img986.svg" alt="${\mathrm{senh}}^{-1} u$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\mathbb{R}$
             -->
            <span class="MATH"><img style="height: 1.64ex; vertical-align: -0.10ex; " src="img/img217.svg" alt="$\mathbb{R}$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\frac{1}{\sqrt{1 + u^{2}}}$
             -->
            <span class="MATH"><img style="height: 3.29ex; vertical-align: -1.31ex; " src="img/img1109.svg" alt="$\frac{1}{\sqrt{1 + u^{2}}}$" loading="lazy"></span> <br>
            <br></td>
            </tr>
            <tr><td class="LEFT"><!-- MATH
             $\cosh^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img1054.svg" alt="$\cosh^{-1} u$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $[1,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img547.svg" alt="$[1,\infty)$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\frac{1}{\sqrt{u^{2} - 1}}$
             -->
            <span class="MATH"><img style="height: 3.29ex; vertical-align: -1.31ex; " src="img/img1110.svg" alt="$\frac{1}{\sqrt{u^{2} - 1}}$" loading="lazy"></span> <br>
            <br></td>
            </tr>
            <tr><td class="LEFT"><!-- MATH
             ${\mathrm {tgh}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.59ex; vertical-align: -0.58ex; " src="img/img1055.svg" alt="${\mathrm {tgh}}^{-1} u$" loading="lazy"></span></td>
            <td class="CENTER"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img614.svg" alt="$(-1,1)$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\frac{1}{1 - u^{2}}$
             -->
            <span class="MATH"><img style="height: 3.05ex; vertical-align: -1.06ex; " src="img/img1111.svg" alt="$\frac{1}{1 - u^{2}}$" loading="lazy"></span> <br>
            <br></td>
            </tr>
            <tr><td class="LEFT"><!-- MATH
             ${\mathrm{ctgh}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.59ex; vertical-align: -0.58ex; " src="img/img1056.svg" alt="${\mathrm{ctgh}}^{-1} u$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $(-\infty,-1) \cup (1,\infty)$
             -->
            <span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img619.svg" alt="$(-\infty,-1) \cup (1,\infty)$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\frac{1}{1 - u^{2}}$
             -->
            <span class="MATH"><img style="height: 3.05ex; vertical-align: -1.06ex; " src="img/img1111.svg" alt="$\frac{1}{1 - u^{2}}$" loading="lazy"></span> <br>
            <br></td>
            </tr>
            <tr><td class="LEFT"><!-- MATH
             ${\mathrm{sech}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img1057.svg" alt="${\mathrm{sech}}^{-1} u$" loading="lazy"></span></td>
            <td class="CENTER"><span class="MATH"><img style="height: 2.30ex; vertical-align: -0.62ex; " src="img/img112.svg" alt="$(0,1)$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\frac{-1}{u \sqrt{1 - u^{2}}}$
             -->
            <span class="MATH"><img style="height: 3.29ex; vertical-align: -1.31ex; " src="img/img1112.svg" alt="$\frac{-1}{u \sqrt{1 - u^{2}}}$" loading="lazy"></span> <br>
            <br></td>
            </tr>
            <tr><td class="LEFT"><!-- MATH
             ${\mathrm{csch}}^{-1} u$
             -->
            <span class="MATH"><img style="height: 2.14ex; vertical-align: -0.13ex; " src="img/img1058.svg" alt="${\mathrm{csch}}^{-1} u$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\mathbb{R}^{*}$
             -->
            <span class="MATH"><img style="height: 1.67ex; vertical-align: -0.10ex; " src="img/img880.svg" alt="$\mathbb{R}^{*}$" loading="lazy"></span></td>
            <td class="CENTER"><!-- MATH
             $\frac{-1}{|u| \sqrt{1+u^{2}}}$
             -->
            <span class="MATH"><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"></span></td>
            </tr>
            </tbody></table>
             <a id="tabderhipinv"></a>
            </div></td></tr>
            </tbody></table>
            </div>
            <br>
            
            <p class=" unidade" id="2P300" title="2P300">
            Note que as derivadas das funções <!-- MATH
             ${\mathrm {tgh}}^{-1}u$
             -->
            <span class="MATH"><img style="height: 2.59ex; vertical-align: -0.58ex; " src="img/img1055.svg" alt="${\mathrm {tgh}}^{-1} u$" loading="lazy"></span> e <!-- MATH
             ${\mathrm{ctgh}}^{-1}u$
             -->
            <span class="MATH"><img style="height: 2.59ex; vertical-align: -0.58ex; " src="img/img1056.svg" alt="${\mathrm{ctgh}}^{-1} u$" loading="lazy"></span> são iguais, porém estão definidas em conjuntos
            disjuntos, isto é, conjuntos que não possuem pontos em comum.
            </p>

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