1. ## hyperbolic functions

Prove that the $\displaystyle sinh^{-1}x = ln(x+\sqrt{x^{2}+1})$

is $\displaystyle sinh^{-1}x = \frac{1}{sinhx}$ ?

if so than could I write

$\displaystyle sinh^{-1}x = \frac{2}{e^{x}-e^{-x}}$

and am not sure how to got from there, any help appreciated.

2. ## Re: hyperbolic functions

Originally Posted by Tweety
Prove that the $\displaystyle sinh^{-1}x = ln(x+\sqrt{x^{2}+1})$
is $\displaystyle sinh^{-1}x = \frac{1}{sinhx}$ ?
First of all $\displaystyle \sinh^{-1}x \not = \frac{1}{\sinh x}$

Many of us hate that notation.
$\displaystyle \sinh^{-1}x$ means the inverse of the hyperbolic-sine function

Find the inverse of $\displaystyle y=\frac{e^x-e^{-x}}{2}$.

3. ## Re: hyperbolic functions

so $\displaystyle sinh^{-1}x = \frac{2}{e^{x}-e^{-x}}$ ?

4. ## Re: hyperbolic functions

Originally Posted by Tweety
so $\displaystyle sinh^{-1}x = \frac{2}{e^{x}-e^{-x}}$ ?
Absolutely not true.

Do you even know what an inverse is?

Do you even know how to find the inverse of a given function?

5. ## Re: hyperbolic functions

I do^^ but I am just a bit confused as to how to do this question.

6. ## Re: hyperbolic functions

Originally Posted by Tweety
I do^^ but I am just a bit confused as to how to do this question.
Solve $\displaystyle x=\frac{e^y-e^{-y}}{2}$ for $\displaystyle y$.

7. ## Re: hyperbolic functions

This might be difficult to see how to solve. You can probably see $\displaystyle e^x=\cosh{x}+\sinh{x}$, but it might not be as easy to see $\displaystyle \cosh^2{x}-\sinh^2{x}=1$.

- Hollywood