Question

The sum of all the solutions of the equation \( \cos \theta \cos \left( \frac{\pi}{3} + \theta \right) \cos \left( \frac{\pi}{3} - \theta \right) = \frac{1}{4} \), for \( \theta \in [0, 6\pi] \) is:

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

Hint:

When solving trigonometric equations involving multiple angles, look for symmetry and periodicity to find all solutions over a given interval.

Answer 1

The Correct Option is B

Starting with the equation given: \[ 2\cos\left(\cos 120^\circ + \cos 2\theta\right) = 1 \] This simplifies as: \[ 2\cos\left(-\frac{1}{2} + 2\cos^2\theta - 1\right) = 1 \] Which further reduces to: \[ 2\cos\left(2\cos^2\theta - \frac{3}{2}\right) = 1 \] Expanding this, we find: \[ 4\cos^3\theta - 3\cos\theta - 1 = 0 \] Solving for \(\theta\), we equate to the general solution of trigonometric equations: \[ 3\theta = 2n\pi { or } \theta = \frac{2n\pi}{3}, \quad n \in \mathbb{Z} \] Ensuring \(2n\) does not exceed 18, we calculate the sum: \[ \sum_{n=1}^9 \frac{2n\pi}{3} = \frac{2\pi}{3} \times \frac{9(9+1)}{2} = 30\pi \]