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..

also note that $x^2 + (iy)^2 = x^2 - y^2$, the hyperbola in real xy plane looks like circle in complex plane and vice versa. — Monkey D. Luffy, May 06 '13 at 10:11

Inceptio · Answer 1 · 2013-05-06T09:45:36.953

2

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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)

edited May 06 '13 at 09:45

answered May 06 '13 at 09:33

Inceptio

7,881

thanks a lot..that was of great help... – Sachin May 06 '13 at 10:05

score 0 · Answer 2 · answered May 06 '13 at 09:24

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.

Trigonometric hyperbolic function - sinhx

2 Answers2