Question

The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2\theta = \cos 2\theta \) and \( 2\cos^2\theta = 3\sin\theta \) is:

• A. \( \frac{\pi}{2} \)
• B. \( 4\pi \)
• C. \( \pi \)
• D. \( \frac{5\pi}{6} \)

Hint:

To solve trigonometric equations involving identities like \( \cos 2\theta \) and \( \sin^2\theta \), - Use standard trigonometric identities and algebraic manipulations to simplify the equations. - Solving step by step and checking for all possible values of \( \theta \) in the given range will help in obtaining the correct sum of solutions.

Answer 1

The Correct Option is C

The given system of trigonometric equations can be solved as follows: 1. From \( 2\sin^2\theta = \cos 2\theta \), using the identity \( \cos 2\theta = 1 - 2\sin^2\theta \), we get: \[ 2\sin^2\theta = 1 - 2\sin^2\theta \quad \Rightarrow \quad 4\sin^2\theta = 1 \quad \Rightarrow \quad \sin^2\theta = \frac{1}{4}. \] This gives \( \sin\theta = \pm \frac{1}{2} \). 2. Substituting these values into the second equation \( 2\cos^2\theta = 3\sin\theta \), we find that \( \theta \) satisfies the range from 0 to \( 2\pi \), yielding the sum of solutions as \( \pi \). Thus, the sum of all values of \( \theta \) is \( \pi \).