# If $\sin^{-1} x=y$ then:

$(A) \quad 0 \leq y \leq \pi \qquad (B) \quad -\frac {\pi} {2} \leq y \leq \frac{\pi}{2}$ $(C) \quad 0 < y < \pi \qquad (B) \quad -\frac{\pi}{2} < y < \frac{\pi}{2}$

Toolbox:
• The range of the principal value of $\sin^{-1}x$ is $\left [ -\large\frac{\pi}{2}, \large\frac{\pi}{2} \right ]$
Since the range of the principal value of $\sin^{-1}x$ is $\left [ -\large\frac{\pi}{2},\large \frac{\pi}{2} \right ]$, if $\sin^{-1}x =y$, then $y$ must be in the range of the principal value.

Therefore, $(B): -\large\frac{\pi}{2} \leq y \leq \large\frac{\pi}{2}$ is the correct answer.

edited Mar 15, 2013