Question

In Problems 1– 8, find the exact value of each expression. Do not use a calculator.

${\sin }^{-1}\left(-\frac{1}{2}\right)$

Short answer

The exact value is $-\frac{\mathrm{\pi }}{6}$.

Step-by-step solution

Step 1. Definition 

The inverse sine function $y={\sin }^{-1}x$means that $x=\sin y$where $-1\le x\le 1$and $-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$.

Step 2. Find the exact value 

Let $\theta ={\sin }^{-1}\left(-\frac{1}{2}\right)$, so using the definition of inverse sine function it can be written as $-\frac{1}{2}=\sin \theta$where $-\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$.

In this interval the angle for which the sine value is $-\frac{1}{2}$is role="math" localid="1646505832673" $-\frac{\mathrm{\pi }}{6}$.

So the value of ${\sin }^{-1}\left(-\frac{1}{2}\right)$is $-\frac{\mathrm{\pi }}{6}$.