# Mathematical pendulum - Arbitrary-amplitude period

1. Oct 15, 2009

### dingo_d

1. The problem statement, all variables and given/known data
Hi! I have a problem with mathematical pendulum. I had to prove that the right expression for the period of the mathematical pendulum is:
$T=\sqrt{\frac{8l}{g}}\int_0^{\theta_0}\frac{d\theta}{\sqrt{\cos \theta- \cos \theta_0}}$
I did that via energy, but now I have to transform that integral with:
$\cos\theta=1-2\sin^2(\theta/2)$ and with this substitution
$\sin x =\sin(\theta/2)/\sin(\theta_0/2)$ so that I could expand that integral into Taylor series.

The attempt at a solution

I've tried with substitution and I get:
$2\sqrt{\frac{l}{g}}\int_0^{\theta}\frac{d\theta}{\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}}$
Now I have seen what the expansion is on wikipedia, but there is no solution to how to get to there. If I try to use the sine substitution, I get stuck at changing the integral variables, and the integral becomes a mess. I saw that the complete elliptic integral of 1st order appears, but I don't know what to do with that.

Can anyone help with this clue? How to get that Taylors expansion?

Thank you!

2. Oct 17, 2009

Anyone?