# Thread: Basic Hyperbolic Simplification b

1. ## Basic Hyperbolic Simplification b

second question:
$\displaystyle cosh^{-1}2=?$
I tried
$\displaystyle \frac{1}{cosh2}=sech2=\frac{2}{e^2+e^{-2}}$

The answer given is $\displaystyle ln(1+\sqrt{3})$
How did they get to a answer w/ natural log?

2. Because $\displaystyle \cosh^{-1}{x} = \ln{(x + \sqrt{x^2 - 1})}$.

3. Unfortunate notation- $\displaystyle f^{-1}(x)$ means the inverse function, not the reciprocal!

Specifically, $\displaystyle cosh^{-1}(2)$ is the value of x such that $\displaystyle cosh(x)= \frac{e^x+ e^{-x}}{2}= 2$. Solve that equation for x: $\displaystyle e^x+ e^{-x}= 4$. Let $\displaystyle y= e^x$ so the equation becomes $\displaystyle y+ y^{-1}= 4$. Multiply both sides by y: $\displaystyle y^2+ 1= 4y$ or $\displaystyle y^2- 4y+ 1= 0$. Solve that equation for y then take the logarithm to find x.

4. And you should note that

$\displaystyle \cosh^{-1}{2} = \ln{(2 + \sqrt{3})}$, not $\displaystyle \ln{(1 + \sqrt{3})}$.

5. Thanks for the clarification on this post and the last post. You guys sure know how to explain things in details better than some professors.