hyperbolic functions

Sep 2008
631
2
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
Aug 2006
22,507
8,664
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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Sep 2008
631
2
so \(\displaystyle sinh^{-1}x = \frac{2}{e^{x}-e^{-x}} \) ?
 

Plato

MHF Helper
Aug 2006
22,507
8,664
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?
 
Sep 2008
631
2
I do^^ but I am just a bit confused as to how to do this question.
 

Plato

MHF Helper
Aug 2006
22,507
8,664
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 \(\displaystyle y\).
 
Mar 2010
1,055
290
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