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The principal value of $\sin -1\left\{\sin \frac{5 \pi}{6}\right\}$ is
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1743 Upvotes
Verified Answer
The correct answer is:
$\frac{\pi}{6}$
$$
\begin{array}{l}
\begin{aligned}
\sin ^{-1}\left\{\sin \frac{5 \pi}{6}\right\}=& \sin ^{-1}\left\{\sin \left(\pi-\frac{\pi}{6}\right)\right\} \\
&=\sin ^{-1}\left\{\sin \frac{\pi}{6}\right\} \\
&[\because \text { Principal value } \in[0, \pi / 2]] \\
=& \frac{\pi}{6}
\end{aligned}
\end{array}
$$
which is the required principal value.
\begin{array}{l}
\begin{aligned}
\sin ^{-1}\left\{\sin \frac{5 \pi}{6}\right\}=& \sin ^{-1}\left\{\sin \left(\pi-\frac{\pi}{6}\right)\right\} \\
&=\sin ^{-1}\left\{\sin \frac{\pi}{6}\right\} \\
&[\because \text { Principal value } \in[0, \pi / 2]] \\
=& \frac{\pi}{6}
\end{aligned}
\end{array}
$$
which is the required principal value.
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