Question

The value of ${\cos }^{-1}\left(\frac{1}{\sqrt{2}}(\cos \frac{7\pi }{5}-\sin \frac{2\pi }{5})\right)$ is

• A. $\frac{23\pi }{20}$
• B. $\frac{13\pi }{20}$
• C. $\frac{3\pi }{20}$
• D. $\frac{17\pi }{20}$

Answer 1

The correct option is D $\frac{17\pi }{20}$

$\frac{1}{\sqrt{2}}(\cos \frac{7\pi }{5}-\sin \frac{2\pi }{5})=\cos \frac{\pi }{4}\cos \frac{7\pi }{5}-\sin \frac{\pi }{4}\sin \frac{2\pi }{5}=\cos \frac{\pi }{4}\cos \frac{7\pi }{5}+\sin \frac{\pi }{4}\sin \frac{7\pi }{5}$
$=\cos (\frac{7\pi }{5}-\frac{\pi }{4})=\cos \left(\frac{23\pi }{20}\right)=\cos (\pi +\frac{3\pi }{20})=-\cos \left(\frac{3\pi }{20}\right)$

$\therefore {\cos }^{-1}\left\{\frac{1}{\sqrt{2}}(\cos \frac{7\pi }{5}-\sin \frac{2\pi }{5})\right\}={\cos }^{-1}(-\cos \left(\frac{3\pi }{20}\right))=\pi -\frac{3\pi }{20}=\frac{17\pi }{20}$
(since $\frac{17\pi }{20}$ lies between $0$ and $\pi$)