Answer
$\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\ x=\displaystyle \frac{2}{3}$
$\displaystyle \cosh x=\frac{\sqrt{13}}{2} \qquad $
$\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\ x=\displaystyle \frac{2}{\sqrt{13}}$
$\displaystyle \tanh x=\frac{3\sqrt{13}}{13} $
$\displaystyle \coth x=\frac{\sqrt{13}}{3}$
Work Step by Step
$\displaystyle \sinh x=\frac{3}{2}$
$\begin{align*}
\cosh^{2}x-\sinh^{2}x&=1\\
\cosh^{2}x&=1+\sinh^{2} \\
\cosh x&=\sqrt{1+(3/2)^{2}} \\
&=\sqrt{13/4} \\\\
& =\displaystyle \frac{\sqrt{13}}{2} \end{align*}$
$\displaystyle \sinh x=\frac{3}{2} \qquad $
$\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}\ x=\displaystyle \frac{1}{\sinh x}=\frac{2}{3}$
$\displaystyle \cosh x=\frac{\sqrt{13}}{2} \qquad $
$\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\ x=\displaystyle \frac{2}{\sqrt{13}}$
$\displaystyle \tanh x=\frac{\sinh x}{\cosh x} =\frac{\frac{3}{2}}{\frac{\sqrt{13}}{2}} = \frac{3}{\sqrt{13}}\cdot\frac{\sqrt{13}}{\sqrt{13}} =\frac{3\sqrt{13}}{13} $
$\displaystyle \coth x=\frac{1}{\tanh x}=\frac{\sqrt{13}}{3}$
