Why does $\sin(x)=\sin(y)$ not equal $\sin^{-1}\left(\sin\left(x\right)\right)=y$?

Question

The equation $\sin(x) = \sin(y)$ does not produce the same output as $y=\sin^{-1}(\sin(x))$. Can anyone explain why this is?

Answer 1

Note that the range $y$ in the explicit function is

$$y=\sin^{-1}(\sin(x))\in \left[-\tfrac\pi2, \tfrac\pi2\right] $$

while the range of $y$ in the implicit function $$\sin y= \sin x,\>\>\>\>y\in (-\infty, \infty)$$