Help solving $f(\sin(x))=\sin(xy)$ for $f(x)$?

Question

I have worked with trigonometric functions at a high school level, but I have not really worked with functional equations or trig functions enough to know how to solve this myself. Is there some rule for this situation? Any hints or solutions would be much appreciated.

Answer 1

If $y$ is an integer, then $\sin (xy)$ is a polynomial in $\sin x$ and $\cos x,$ so that if we set $z=\sin x,$ then we obtain that $f(z)$ is a sum of powers of $z.$ That is, a linear combination of expressions of the form $z^m(\sqrt{1-z^2})^n$, where $m,n$ are non-negative integers.

Answer 2

$$ f( \sin x)= \sin (x y) $$

In a bounded argument:

$$ x\rightarrow \arcsin x $$ $$ f(x)= \sin (y\cdot \arcsin x ) $$

$f(x) $ cannot be solved out analytically from implicit form.