# If tanh x =7/25 how do you find the values of the other hyperbolic functions at x?

Dec 25, 2016

#### Answer:

$\sinh x = \frac{7}{24}$, $\cosh x = \frac{25}{24}$, $\tanh x = \frac{7}{25}$
$\coth x = \frac{25}{7}$, $\sech x = \frac{24}{25}$, $\csch x = \frac{24}{7}$

#### Explanation:

Apart from the usual relations that

$\tanh x = \sinh \frac{x}{\cosh} x$, $\coth x = \cosh \frac{x}{\sinh} x$, $\sech x = \frac{1}{\cosh} x$ and $\csch x = \frac{1}{\sinh} x$,

some other relations we can use are

${\cosh}^{2} x - {\sinh}^{2} x = 1$ and ${\tanh}^{2} x + {\sech}^{2} x = 1$

Hence, as $\tanh x = \frac{7}{25}$, $\coth x = \frac{25}{7}$

$\sech x = \sqrt{1 - {7}^{2} / {25}^{2}} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}$

$\cosh x = \frac{25}{24}$, $\sinh x = \tanh x \times \cosh x = \frac{7}{25} \times \frac{25}{24} = \frac{7}{24}$ and

$\csch x = \frac{24}{7}$