Question

Find the values of ${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$ is equal to

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

Answer 1

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

Let $y={\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$

$\cos y=\cos \left(\frac{7\pi }{6}\right)$

$\cos y=\cos \left({210}^o\right)$

We know that range of principal values is $\cos$ is $(0,\pi )$

$i.e(0^o,{180}^o)$

Hence $4={210}^o$ not possible

Now,

$\cos y=\cos \left({210}^o\right)$

$\cos y=\cos (360-{150}^o)$

$\cos y=\cos \left({150}^o\right)$

$\cos y=\cos \left(\frac{5\pi }{5}\right)$

$4=\frac{5\pi }{6}$

Which is in range of principal values of ${\cos }^{-1}$ i.e $(0,\pi )$

$\therefore y=\frac{5\pi }{6}$