$\arcsin(x)$ without use of itself and $\sin^{-1}(x)$

Question

I need to calculate $\arcsin(x)$ but I can only use $\sin(x)$, $\cos(x)$, basic operators, powers and roots. $\sin^{-1}(x)$ is not available. How can I do this?

In short, I need another way to calculate this: $$2\arcsin\left(\frac{r}{d}\right)$$ without using $\arcsin$.

Thanks for any help.

Answer 1

$$\arcsin x = \sum_{n\ge 0} \frac{(2n)!}{2^{2n}(n!)^2}\frac{x^{2n+1}}{2n+1}$$

Only addition, multiplication, and division required!

If you only need an approximation to arcsine (as it appears, from the comments), then you don't need the whole series, just truncate after some number of terms to get an approximation.

$$\arcsin x = x + \frac{x^3}{6}+\frac{3x^5}{40}+\frac{5x^7}{112}+\cdots$$

Answer 2

You may also use Newton's method with starting point $z_0=\frac{3x}{2+\sqrt{1-x^2}}$ (provided by the Shafer-Fink inequality) - the sequence defined by $$z_{n+1} = z_n-\frac{\sin(z_n)-x}{\cos(z_n)} $$ converges to $\arcsin(x)$ pretty fast.