# hyperbolic functions

#### Tweety

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.

#### Plato

MHF Helper
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 $$\displaystyle y=\frac{e^x-e^{-x}}{2}$$.

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2 people

#### Tweety

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

#### Plato

MHF Helper
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?

#### Tweety

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

#### Plato

MHF Helper
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 $$\displaystyle y$$.

#### hollywood

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