Rephrasing the question:

What are the functions g : [-1, 1] to R such that

f(g(x)) = x (or sin(g(x)) = x)?

g = sin^-1(x) is one solution

What about g(x) = sin^-1(x) + 2*pi?

What about g(x) = sin^-1(x) + 4*pi?

What about g(x) = -sin^-1(x) + pi?

What about g(x) = -sin^-1(x) + 3*pi?

Also, the function does not even need to be continuous. What about the function

g(x) = sin^-1(x) + 2*pi for 0 <= x <= 1

and g(x) = sin^-1(x) for -1 <= x < 0

Can you think of the most general form for g(x)?