Solving short trigo equation with sine - need some help! From the relation $M=E-\epsilon\cdot\sin(E)$, I need to find the value of E, knowing the two other parameters. How should I go about this?
This is part of a computation which will be done quite a number of times per second. I hope there's a quick way to get E out of this equation.
Thank you very much,
MJ

I assume $\epsilon$ is a small quantity and propose one of the following:
(a) Write your equation in the form $E=M+\epsilon \sin(E)=: f(E)$ and consider this as a fixed point problem for the function $f$. Starting with $E_0:=M$ compute numerically successive iterates $E_{n+1}:=f(E_n)$; these will converge to the desired solution of the given equation.
(b) $E$ depends in an analytic way on the parameter $\epsilon$. Make the "Ansatz" $E:=M +\sum_{k=1}^\infty a_k \epsilon^k$ and determine the coefficients $a_k$ recursively. You will find $a_1=\sin(M)$, $a_2=\cos(M)\sin(M)$ and so on.
