Question

Find the number of roots of the equation $5\cos 2\theta +2{\cos }^2\frac{\theta }{2}+1=0$ in $[0,\pi ]$

Answer 1

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

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

$\Rightarrow 10{\cos }^2\theta -5+1+\cos \theta +1=0$

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

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

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

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

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

If $\cos \theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$

$\cos \theta =\frac{-3}{5}\Rightarrow \theta =\pi -{\cos }^{-1}\frac{3}{5}$

$\therefore$ there are $2$ roots of the above equation

$\{\frac{\pi }{3},\pi -{\cos }^{-1}\frac{3}{5}\}$