Graph of the Inverse Sine Function

$\sin^{-1}:[-1,1]\to\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$ receives $x\in[-1,1]$ and returns $y\in\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$. As $x$ increases from $-1$ to $1$, $y$ increases from $-\tfrac{\pi}2$ to $\tfrac{\pi}2$; connecting the plotted points $(x,y)$ with a smooth curve produces Fig. 4.6.

Equivalently, Fig. 4.6 is obtained by reflecting the restricted sine curve of Fig. 4.7 (on $\left[-\tfrac{\pi}2,\tfrac{\pi}2\right]$) in the line $y=x$ — interchanging its $x$- and $y$-axes, as shown against the reflection line in Fig. 4.8. The graph passes through the origin and is symmetric about the origin, confirming once more that $y=\sin^{-1}x$ is an odd function. …