Question

The value of ${\cos }^{-1}\left(\cos \left(\frac{5\pi }{3}\right)\right)+{\sin }^{-1}\left(\sin \left(\frac{5\pi }{3}\right)\right)$

• A. $0$
• B. $\frac{\pi }{2}$
• C. $\frac{2\pi }{3}$
• D. $\frac{10\pi }{3}$

Answer 1

The correct option is A

$0$

Explanation for correct option

Given trigonometric function is ${\cos }^{-1}\left(\cos \left(\frac{5\pi }{3}\right)\right)+{\sin }^{-1}\left(\sin \left(\frac{5\pi }{3}\right)\right)$

$$\begin{gathered} {\cos }^{-1}\left(\cos \left(\frac{5\pi }{3}\right)\right)+{\sin }^{-1}\left(\sin \left(\frac{5\pi }{3}\right)\right) \\ ={\cos }^{-1}\left(\cos \left(2\pi -\frac{\pi }{3}\right)\right)+{\sin }^{-1}\left(\sin \left(2\pi -\frac{\pi }{3}\right)\right) \\ ={\cos }^{-1}\left(\cos \left(\frac{\pi }{3}\right)\right)+{\sin }^{-1}\left(-\sin \left(\frac{\pi }{3}\right)\right) \\ ={\cos }^{-1}\left(\cos \left(\frac{\pi }{3}\right)\right)-{\sin }^{-1}\left(\sin \left(\frac{\pi }{3}\right)\right) \\ =\frac{\pi }{3}-\frac{\pi }{3} \\ =0 \end{gathered}$$

Hence the correct option is $\mathrm{Option}\left(\mathrm{A}\right)$