# If tanh(x) = 4/5, how do you find the values of the other hyperbolic functions at x?

Nov 20, 2017

$\sinh x = \frac{4}{3}$, $\cosh x = \frac{5}{3}$, $\tanh x = \frac{4}{5}$
$\coth x = \frac{5}{4}$, $\sech x = \frac{3}{5}$, $\coth x = \frac{5}{4}$

#### Explanation:

As ${\tanh}^{2} x + {\sech}^{2} x = 1$ and $\tanh x = \frac{4}{5}$

hence $\sech x = \sqrt{1 - {\left(\frac{4}{5}\right)}^{2}} = \frac{3}{5}$

and $\cosh x = \frac{1}{\sech} x = \frac{5}{3}$

$\coth x = \frac{1}{\tanh} x = \frac{5}{4}$

$\sinh x = \tanh x \cdot \cosh x = \frac{4}{5} \cdot \frac{5}{3} = \frac{4}{3}$

$\csch x = \frac{1}{\sinh} x = \frac{3}{4}$