[Math] Given $y=\arccos(x)$ find $\arcsin(x)$ in terms of y

Question

Given that $y = \arccos x$, $ – 1 \le x \le 1\,and\,0 \le y \le \pi $, express $\arcsin x$ in terms of y.

The best I know how to do this is is:

$$\eqalign{
& \cos y = x \cr
& {\cos ^2}y + {\sin ^2}y = 1 \cr
& {\sin ^2}y = 1 – {\cos ^2}y \cr
& \sin y = \sqrt {1 – {x^2}} \cr
& \arcsin \left( {\sqrt {1 – {x^2}} } \right) = y \cr} $$

However this isn't what is asked for, How do I go about getting things in terms of y?

Thanks

Answer 1

HINT: $$x=\cos y=\sin\left(\frac{\pi }{2}-y\right)$$