Finding values of hyperbolic functions

Question

Struggling in Calc2, the question gives a value of sinh x = -3/4 and asking me to find the values of the remaining five hyperbolic functions. Can anybody help me as to how to approach this problem? I have the answers from the back of the book but I cannot understand how to answer it. Any help would be appreciated.

Answer 1

Use the identity $\cosh^2x-\sinh^2x \equiv 1$.

If $\sinh x = \frac{3}{4}$ then $$\cosh^2x - \left(\frac{3}{4}\right)^{\! 2} = 1$$ $$\cosh^2x - \left(\frac{9}{16}\right) = 1$$ $$\cosh^2x =\left(\frac{25}{16}\right)$$

It follows that $\cosh x = \pm\frac{1}{2}5/4$. Since $\cosh x \ge 1$ for all $x \in \mathbb{R}$ we have $\cosh x = \frac{1}{2}\sqrt{5/4}$.

Then, we find $\tanh(x)$ thanks to:

$\tanh(x)=\left(\frac{sinh(x)}{cosh(x)}\right)$.

And $sech(x)$ thanks to:

$sech(x)=\left(\frac{1}{cosh(x)}\right)$.

Again $cosech(x)$ with:

$cosech(x)=\left(\frac{1}{sinh(x)}\right)$.