Question

Solve: $5\cos 2\theta +2{\cos }^2\left(\frac{1}{2}\theta \right)+1=0,-\pi <\theta <\pi$.

Answer 1

Given, $5(2{\cos }^2\theta -1)+(1+\cos \theta )+1=0$

$10{\cos }^2\theta +\cos \theta -3=0$

$10{\cos }^2\theta +6\cos \theta -5\cos \theta -3=0$

$(5\cos \theta +3)(2\cos \theta -1)=0$

$\therefore \theta =\pi /3,-\pi /3,\pi -{\cos }^{-1}\left(3/5\right)$ $\because -\pi <\theta <\pi$.