## Precalculus (6th Edition)

$\frac{\pi}{4}$
We know that $y=\arcsin x$ means $y$ is the value in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ whose sine is $x$. The solutions to $\sin y=\frac{\sqrt{2}}{2}$ are $y=..., -\frac{7\pi}{4}, -\frac{5\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}, \frac{9\pi}{4}, \frac{11\pi}{4}, ...$. The only solution in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is $y=\frac{\pi}{4}$. Therefore, the solution to $y=\arcsin \frac{\sqrt{2}}{2}$ is $y=\frac{\pi}{4}$.