hyperbolic functions

**Tweety** · October 25th 2012, 07:49 AM

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

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

if so than could I write

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

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

**Plato** · October 25th 2012, 08:17 AM

Originally Posted by Tweety

Prove that the $sinh^{-1}x = ln(x+\sqrt{x^{2}+1})$
is $sinh^{-1}x = \frac{1}{sinhx}$ ?

First of all $\sinh^{-1}x \not = \frac{1}{\sinh x}$

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

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

**Tweety** · October 25th 2012, 09:48 AM

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

**Plato** · October 25th 2012, 09:55 AM

Originally Posted by Tweety

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

Absolutely not true.

Do you even know what an inverse is?

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

**Tweety** · October 25th 2012, 10:48 AM

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

**Plato** · October 25th 2012, 10:56 AM

Originally Posted by Tweety

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

Solve $x=\frac{e^y-e^{-y}}{2}$ for $y$ .

**hollywood** · October 25th 2012, 08:17 PM

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

- Hollywood

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