# If tanhx=12/13, how do you find the values of the other hyperbolic functions at x?

Sep 18, 2016

$\sinh x = \pm \frac{12}{5}$, $\cosh x = \pm \frac{13}{5}$, $\coth x = \frac{13}{12}$, $\sech x = \pm \frac{5}{13}$ and $\csch x = \pm \frac{5}{12}$

#### Explanation:

We can use relations between hyperbolic functions to find them.

As ${\sech}^{2} x = 1 - {\tanh}^{2} x$

${\sech}^{2} x = 1 - {\left(\frac{12}{13}\right)}^{2} = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169}$

or $\sech x = \pm \frac{5}{13}$

$\cosh x = \frac{1}{\sech} x = \pm \frac{13}{5}$

Hence $\sinh x = \tanh x \times \cosh x = \pm \frac{12}{13} \times \frac{13}{5} = \pm \frac{12}{5}$

and $\csch x = \frac{1}{\sinh} x = \pm \frac{5}{12}$

$\coth x = \frac{1}{\tanh} x = \frac{13}{12}$