Does $\sin^{-1}$ abide with the trig exponent notation?

Question

So I know that the difference between "$\sin^a$" and "$\sin^a$" is that "$\sin^a$" means "$(\sin)^a$" and "$\sin^a$" means "$\sin(^a)$", but does this work for -1? I know notation for the inverse of "$\sin$" is "$\sin^{-1}$", but this would mean "$(\sin)^{-1}$", which is not true. For example, "$\sin^{-1}1$" is "$\frac{\pi}{2}$" but "$\left(\sin\frac{\pi}{2}\right)^{-1}$" is not the same. (In conclusion, "$\sin^{-1}$" is not the same as "$(\sin)^{-1}$".) Why not?

Answer 1

It's the same with every other function. $f^{-1}(x)$ means the inverse of $f$, and $f^n(x)$ means $(f(x))^n$ for all $n \neq 0$. It's the most despised notation in math.