### Instruction: Compute $\arcsin (-1).$  Express your answer in radians.### Response: The arcsine function is the inverse of the sine function, so I want to find the angle $\theta$ such that $\sin \theta = -1.$
I know the sine function is periodic with period $2\pi$, so there could be infinitely many such angles, but I only need one that lies in the domain of the arcsine function, which is $[-\frac{\pi}{2}, \frac{\pi}{2}].$
I also know the sine function is an odd function, so $\sin (-\theta) = -\sin \theta.$
That means if $\theta$ is a solution, so is $-\theta.$
I recall the special angles for the sine function, and I remember that $\sin \frac{\pi}{2} = 1$ and $\sin -\frac{\pi}{2} = -1.$
Therefore, $\theta = -\frac{\pi}{2}$ is a solution that satisfies both the equation and the domain.
Hence, $\arcsin (-1) = -\frac{\pi}{2}.$