Question

$\cos \left(\frac{1}{2}{\cos }^{-1}\left[\cos \right(-14\frac{\pi }{5}\left)\right]\right)=$

• A. $\cos \left(\frac{9\pi }{5}\right)$
• B. $\sin \left(\frac{-\pi }{5}\right)$
• C. $\cos \left(\frac{2\pi }{5}\right)$
• D. $-\cos \left(\frac{3\pi }{5}\right)$

Answer 1

The correct option is C $\cos \left(\frac{2\pi }{5}\right)$

Let us be very careful about the interval in which the angle belongs to
$\theta =-\frac{14\pi }{5}=-2.8\pi$
$\therefore -3\pi$ < $\theta$ < $-2\pi$
i.e., [2(-1)-1] $\pi$ < $\theta$ < $-2\pi$.
$\therefore {\cos }^{-1}\left(\cos \theta \right)=2(-1)\pi -\theta =-2\pi -\theta =-2\pi +\frac{14\pi }{5}=\frac{4\pi }{5}\therefore \cos (\frac{1}{2}{\cos }^{-1}.\cos (-\frac{14\pi }{5}))=\cos (\frac{1}{2}.\frac{4\pi }{5})=\cos \left(\frac{2\pi }{5}\right)$