Simplifying ArcSine Function

Question

I was wondering if there is a nice formula (or approximation) for $\arcsin(x)$ which is defined $[-1,1]$?

Answer 1

As noted there is no way to explicitly express $\arcsin(x)$ in terms of e.g. the $\sin,\cos,\tan,\exp,\log$ functions. After all, if there were, why would we need to invent the new notation $\arcsin$? There are however nice approximations to the function. The most obvious one which comes to mind is the Taylor approximation. We have the Taylor series expansion $$\arcsin(x)=\sum_{k=0}^\infty\frac{(2k-1)!!}{(2k)!!}\frac{x^{2k+1}}{2k+1}.$$ If we truncate the sum by discarding the higher order terms, we end up with decent polynomial approximations to $\arcsin(x)$ on the interval. Many other approximations exist, for example the Padé approximant, which gives for example $$\arcsin(x)\approx\frac{x-(1709/2196)x^3+(69049/922320)x^5}{1-(2075/2196)x^2+(1075/6832)x^4}.$$ These are not unique to $\arcsin$, of course, and can be applied to any sufficiently nice function.