$\sin^{-1} x$, also written $\arcsin x$ and read aloud arcsine of x, means the angle whose sine is x. The $-1$ on a function's name marks the inverse function, the way $f^{-1}$ undoes $f$; it is NOT an exponent. The reciprocal $\dfrac{1}{\sin x}$ has its own name, $\csc x$, and the two are never equal. Saying arcsine out loud is the cheapest insurance in this chapter.

Sine repeats: $\sin\dfrac{\pi}{6}$, $\sin\dfrac{5\pi}{6}$ and $\sin\dfrac{13\pi}{6}$ are all $\tfrac{1}{2}$. A function may return only ONE output, so each inverse is restricted to a stretch where the original never repeats a value, its principal range: $\arcsin$ answers from $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$, $\arccos$ from $[0, \pi]$, $\arctan$ from $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$, open at the ends because tangent is undefined there. So $\arcsin\tfrac{1}{2} = \dfrac{\pi}{6}$, read aloud arcsine of one half equals pi over six, and never $\dfrac{5\pi}{6}$, however true it is that its sine is one half. The domains follow from what the original can produce: $\arcsin$ and $\arccos$ accept only $[-1, 1]$, because a sine or cosine never leaves it; $\arctan$ accepts any real number.