This is a good question! Trig functions and inverse trig functions are tricky. In fact, IEEE has published standards for how they should be computed. (EDIT: Clause 9 of IEEE 754-2008, the floating point standard, recommends but does not require the implementation of trigonometric functions.)
In the following paper, a method is described for computing $\arcsin x$ when $0 < x < 1$: Paper.
In essence, they use the trig identity
$$\arcsin N = \arctan \frac{N}{\sqrt{1-N^2}}.$$
Of course, this just raises the question: how do you compute $\arctan x$?
Old algorithms used something called CORDIC.
Modern computers have sufficient memory and speed that they can construct lookup tables and interpolate. However, CORDIC is still finding use in things like FPGAs. I don't know offhand what the standard algorithm is, currently, but I'm willing to bet your calculator either uses CORDIC or interpolated lookups.