If coshx = (5/3) and x>0, how do you find the values of other hyperbolic functions at x?

Jul 24, 2016

$\sinh x = \pm \frac{4}{3}$, $\sech x = \frac{3}{5}$, $\csch x = \pm \frac{3}{4}$, $\tanh x = \pm \frac{4}{5}$ and $\coth x = \pm \frac{5}{4}$

Explanation:

Relation between $\sinh x x$ and $\cosh x$ is given by

${\cosh}^{2} x - {\sinh}^{2} x = 1$ and hence

$\sinh x = \sqrt{{\cosh}^{2} x - 1} = \sqrt{{\left(\frac{5}{3}\right)}^{2} - 1}$

= $\sqrt{\frac{25}{9} - 1} = \sqrt{\frac{16}{9}} = \pm \frac{4}{3}$

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

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

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

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