Question
The principal value of $${\cos ^{ - 1}}\left( { - \sin \frac{{7\pi }}{6}} \right)\,$$ is
A.
$${\frac{{5\pi }}{3}}$$
B.
$${\frac{{7\pi }}{6}}$$
C.
$${\frac{{\pi }}{3}}$$
D.
None of these
Answer :
$${\frac{{\pi }}{3}}$$
Solution :
$$\eqalign{
& {\cos ^{ - 1}}\left( { - \sin \frac{{7\pi }}{6}} \right) = {\cos ^{ - 1}}\left\{ {\cos \left( {\frac{\pi }{2} + \frac{{7\pi }}{6}} \right)} \right\} \cr
& {\cos ^{ - 1}}\left( { - \sin \frac{{7\pi }}{6}} \right) = {\cos ^{ - 1}}\left( {\cos \frac{{5\pi }}{3}} \right) = {\cos ^{ - 1}}\left\{ {\cos \left( {2\pi - \frac{{5\pi }}{3}} \right)} \right\} \cr
& {\cos ^{ - 1}}\left( { - \sin \frac{{7\pi }}{6}} \right) = {\cos ^{ - 1}}\left( {\cos \frac{\pi }{3}} \right) = \frac{\pi }{3}. \cr} $$
Remember, $${\cos^{ - 1}}\left( {\cos x} \right) = x\,\,{\text{if }}0 \leqslant x \leqslant \pi .$$