Question

The sum of all the solutions of the equation $\cos \theta \cos (\frac{\pi }{3}+\theta )\cos (\frac{\pi }{3}-\theta )=\frac{1}{4},\theta ϵ[0,6\pi ]$

• A. $15\pi$
• B. $30\pi$
• C. $\frac{100\pi }{3}$
• D. None of these

Answer 1

Answer: (B)

$30\pi$

$\cos \theta .\cos (\frac{\pi }{3}-\theta ).\cos (\frac{\pi }{3}+\theta )=\frac{1}{4}$

$\Rightarrow \cos \theta .(\frac{1}{4}{\cos }^2\theta -\frac{3}{4}{\sin }^2\theta )=\frac{1}{4}$
$\Rightarrow \cos \theta .(4{\cos }^2\theta -3)=1\Rightarrow 4{\cos }^3\theta -3\cos \theta =1$
$\Rightarrow \cos 3\theta =1$
Therefore general solution is,
$3\theta =2n\pi \pm 0\Rightarrow \theta =\frac{2n\pi }{3}$, where n is any integer
Thus solutions in the given interval are,
$\theta =0,\frac{2\pi }{3},\frac{4\pi }{3},2\pi ,\frac{8\pi }{3},\frac{10\pi }{3},4\pi ,\frac{14\pi }{3},\frac{16\pi }{3},6\pi$
Sum of these solution is $30\pi$
Hence, option 'B' is correct.