1. ## Hyperbolic function identity

Express $4\cosh x+5\sinh x$ in the form $r\sinh (x+y)$ giving the values of $r$ and $\tanh y$.

My problem is I don't know which identities to use to change forms, I've recently started on this hyperbolic functions and am not very familiar with them yet.
Thanks!

2. Hello, arze!

We need:

. . $\sinh(A + B) \:=\:\sinh A\cosh B + \cosh A\sinh B$

. . $\tanh x \:=\:\dfrac{\sinh x}{\cosh x}$

$\text{Express }4\cosh x+5\sinh x\text{ in the form }r\sinh (x+y)$

$\text{Give the values of }r\text{ and }\tanh y$

Let: . $Z \;=\;4\cosh x + 5\sinh x$

Divide by 3: . $\dfrac{Z}{3} \;=\;\frac{4}{3}\cosh x + \frac{5}{3}\sinh x$ .[1]

Let: . $\sinh y = \frac{4}{3},\;\cosh y = \frac{5}{3}$

Substitute into [1]: . $\dfrac{Z}{3} \;=\;\cosh y \sinh x + \cosh y \sinh x$

. . . . And we have: . $\dfrac{Z}{3} \;=\;\sinh(x + y)$

Hence: . $Z \;=\;3\sinh(x + y)\;\text{ where }y \:=\:\sinh^{-1}\left(\frac{4}{3}\right)$

Therefore: . $r \:=\:3$

. . . and: . $\tanh y \:=\:\dfrac{\sinh y}{\cosh y} \:=\:\dfrac{\frac{4}{3}}{\frac{5}{3}} \:=\:\dfrac{4}{5}$