Question

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

• A. $\frac{7\mathrm{\pi }}{6}$
• B. $\frac{5\mathrm{\pi }}{6}$
• C. $\frac{\mathrm{\pi }}{6}$
• D. None of these.

Answer 1

The correct option is B

$\frac{5\mathrm{\pi }}{6}$

Finding the value simplifying the given function by Trigonometric identities:

$\begin{array}{rcl}& & {\cos }^{-1}\left(\cos \left(\frac{7\mathrm{\pi }}{6}\right)\right)\\ & =& {\cos }^{-1}\left(\cos \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{6}\right)\right)\mathbf{\{}\mathbf{\because }\mathit{c}\mathit{o}\mathit{s}\left(\pi +\theta \right)\mathbf{=}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{\}}\\ & =& {\cos }^{-1}\left(-\cos \left(\frac{\mathrm{\pi }}{6}\right)\right)\\ & =& {\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\{\because {\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{-}\mathit{\theta }\mathbf{)}\mathbf{=}\mathbf{\pi }\mathbf{-}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\mathbf{\theta }\mathbf{\}}\\ & =& \mathrm{\pi }-{\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)\\ & =& \mathrm{\pi }-\frac{\mathrm{\pi }}{6}\\ & =& \frac{5\mathrm{\pi }}{6}\end{array}$

Therefore, Option $\left(B\right)$ is correct.