Question

The value of ${\cos }^{-1}\left(\cos \frac{5\mathrm{\pi }}{3}\right)+{\sin }^{-1}\left(\sin \frac{5\mathrm{\pi }}{3}\right)$ is
(a) $\frac{\mathrm{\pi }}{2}$

(b) $\frac{5\mathrm{\pi }}{3}$

(c) $\frac{10\mathrm{\pi }}{3}$

(d) 0

Answer 1

(d) 0

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