Question

If $5 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1-0$, when $(0<\theta<\pi)$, then the values of $\theta$ are

• A. $\frac{\pi}{3} \pm \pi$
• B. $\frac{\pi}{3}, \cos ^{-1}\left(\frac{3}{5}\right)$
• C. $\cos ^{-1}\left(\frac{3}{5}\right) \pm \pi$
• D. $\frac{\pi}{3}, \pi-\cos ^{-1}\left(\frac{3}{5}\right)$

Answer 1

Answer: (D)

Given equation is
$5 \cos 2 \theta+2 \cos ^2 \frac{\theta}{2}+1=0$
$\Rightarrow 5\left(2 \cos ^2 \theta-1\right)+1+\cos \theta+1=0 $
$\Rightarrow 10 \cos ^2 \theta+\cos \theta-3=0$
$\Rightarrow (2 \cos \theta-1)(5 \cos \theta+3)=0$
$\Rightarrow \cos \theta=\frac{1}{2} $
or $\cos \theta=-\frac{3}{5}$
$\Rightarrow \theta= \frac{\pi}{3}$ or $\theta = \pi - \cos ^{-1}\left(\frac{3}{5}\right)$