Question

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

Answer 1

$5\cos 2\theta +2{\cos }^2\frac{\theta }{2}+1=0$

We know that $\cos 2\theta =2{\cos }^2\theta -1$

$\cos \theta =2co{s}^2\frac{\theta }{2}-1\Rightarrow 2{\cos }^2\frac{\theta }{2}=\cos \theta +1$

Substituting these vlaues in the given equation, we get

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

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

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

$\Rightarrow \cos \theta =\frac{-3}{5}$ and $\cos \theta =\frac{1}{2}$

$\Rightarrow \theta =\frac{-\pi }{3},\frac{\pi }{3},{\cos }^{-1}\left(\frac{-3}{5}\right)$