2.2 Definitions and Graphs of Inverse Functions

Let $y = \sin^{-1}(\frac{1}{\sqrt{2}})$. Then $\sin y = \frac{1}{\sqrt{2}}$. Since the principal branch of $\sin^{-1}$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, we look for an angle in this interval whose sine is positive. The angle is in the 1st quadrant. We know $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$. Therefore, the principal value is $\frac{\pi}{4}$.