Trigonometric hyperbolic function - sinhx

Question

It is given that $\sinh x = \frac{e^x-e^{-x}}{2}$ which is given in various sources, however it has not been explained diagramatically or I am unable to get the derivation of these functions.

So, I request you to please explain me about this relation , I will be greatful to you.

Else refer me any source book for this, where this has been explained diagramatically and derivation is also available.

Thanks again..

Answer 1

Given any function $f(x)$, there are unique functions $g(x)$ and $h(x)$ such that $g(x)$ is even (that is, $g(-x)=g(x)$), $h(x)$ is odd (that is, $h(-x)=-h(x)$), and $f=g+h$.

If $f(x)=e^x$, then $g(x)$ is the hyperbolic cosine, and $h(x)$ is the hyperbolic sine.

Answer 2

There's complete proof over here and here. You will see that $\sinh \alpha$ is vertical component and $\cosh \alpha$ is the horizontal component. (As the figure also speaks)