The function $\sin:[-\pi/2,\pi/2]\rightarrow[-1,1]$ has an inverse function $\sin^{-1}:[-1,1]\rightarrow[-\pi/2,\pi/2]$.

We cut down the codomain to the range $[-1,1]$, so that $\sin$ takes all the values in the codomain. Then we cut down the domain to $[-\pi/2,\pi/2]$ so as to make the modified function strictly increasing. Then we can apply Proposition 2.32 to obtain the inverse function.