How can I get a "better small angle approximation"? Is there a way to improve on the small angle approximation so that I get a more accurate answer. Is there any way to use a higher order series for the $\sin$ function or to use some sort of series approximation to the elliptical solution to the equation?
For reference, the equation is $$y'' = -\sin(y).$$

If you use the third order Taylor approximation and solve the equation with constants $c_1$ and $c_2$ you can find a solution with the Elliptic F function rather than amplitude. For convience, let $c_3 = \sqrt{9-3c_1}$.
$$-\frac{6 \left(2 c_3-y^2+6\right) \left(c_3+y^2-6\right) F\left(i \sinh ^{-1}\left(y\sqrt{\frac{1}{2 c_3-6}} \right)|\frac{3-c_3}{c_3+3}\right){}^2}{\left(c_3+3\right) \left(12 c_1+y^4-12 y^2\right)}=\left(c_2+x\right){}^2$$
With a fifth order approximation the solution is so bad that I am not going to post it here. My advice is to just look for another approximation for $\sin$ rather than Taylor series.
This is just too long for the comments so don't bother up/downvoting this post. It's only an answer if the F function is sufficiently nicer than the amplitude function.
