Question

Find the exact value of${\sin }^{-1}\left(-\frac{1}{2}\right)$.

Short answer

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

Step-by-step solution

Step 1. Given data

The given expression is

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

Step 2. Expression value

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

then role="math" localid="1646857520718" $-\frac{1}{2}=\sin \theta$

In interval role="math" $\frac{-\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

role="math" localid="1646857602950" $$\begin{gathered} \sin \theta =-\frac{1}{2} \\ \sin \theta =\sin \left(\frac{-\mathrm{\pi }}{6}\right) \\ \theta =\frac{-\mathrm{\pi }}{6} \end{gathered}$$

Sorole="math" localid="1646857643826" ${\sin }^{-1}\left(-\frac{1}{2}\right)=\frac{-\mathrm{\pi }}{6}$