Sketching the function $y=\sin^{-1}\sin x$ for $x\in\mathbb{R}$

 YouTube Video Link:  YOUTUBE CHANNEL

Question: Sketch $y=\sin^{-1}\sin x$ for $x\in\mathbb{R}$.

Basically you work it out for $-\pi/2 \le x \le \pi/2$ first, which is fairly easy just using the definition of the inverse function. Then you can use symmetry to guess the rest. Below we did work out the details for  $\pi/2 \le x \le 3\pi/2$ and since the function has period $2\pi$ we know the rest of the graph.

The YouTube video talks through the basic step.

Below is a screen shot from the YouTube video for this problem.