1. ## Hyperbolic function identity

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

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:

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

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

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

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

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

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

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

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

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

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

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

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