Question

The value of the expression :
${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$ is equal to

• A. $\frac{\pi }{3}$
• B. $\frac{5\pi }{6}$
• C. $\frac{7\pi }{6}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is B $\frac{5\pi }{6}$

Given: ${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$
We know that the range of the principal value branch of ${\cos }^{-1}$ is $[0,\pi ]$.
$\therefore {\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)={\cos }^{-1}\left(\cos (\pi +\frac{\pi }{6})\right)$
$={\cos }^{-1}(-\cos \left(\frac{\pi }{6}\right))$
$\because {\cos }^{-1}(-x)=\pi -{\cos }^{-1}x$,
$\Rightarrow {\cos }^{-1}(-\cos \left(\frac{\pi }{6}\right))=\pi -{\cos }^{-1}\left(\cos \left(\frac{\pi }{6}\right)\right)$
i.e., ${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)=\pi -\frac{\pi }{6}=\frac{5\pi }{6}$
Hence, the correct option is $\left(B\right).$