Question

The value of $cos\left(\frac{1}{2}co{s}^{-1}\left[cos\right(-14\frac{\pi }{5}\left)\right]\right)$

• 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 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] $\pi$ < $\theta$ < $-2\pi$.
$\therefore co{s}^{-1}\left(cos\theta \right)=2(-1)\pi -\theta =-2\pi -\theta =-2\pi +\frac{14\pi }{5}=\frac{4\pi }{5}\therefore cos(\frac{1}{2}co{s}^{-1}.cos(-\frac{14\pi }{5}))=cos(\frac{1}{2}.\frac{4\pi }{5})=cos\left(\frac{2\pi }{5}\right)$