For $-1 \le x \le 1$, what is $\sin\left(\cos^{-1} x\right)$?

Explanation

1. 01

   Let $\theta = \cos^{-1} x$, so $\cos \theta = x$ and $\theta$ is in $[0, \pi]$, where $\sin \theta \ge 0$.

2. 02

   $\sin \theta = \sqrt{1 - \cos^{2} \theta} = \sqrt{1 - x^{2}}$, the positive root.