Question

If $5\cos 2\theta +2{\cos }^2\frac{\theta }{2}=-1,$ 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

The correct option is D $\frac{\pi }{3},\pi -{\cos }^{-1}\left(\frac{3}{5}\right)$

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