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 solution to the system of trigonometric equations is derived as follows:1. The first equation, \( 2\sin^2\theta = \cos 2\theta \), is transformed using the identity \( \cos 2\theta = 1 - 2\sin^2\theta \). This yields: \[ 2\sin^2\theta = 1 - 2\sin^2\theta \quad \Rightarrow \quad 4\sin^2\theta = 1 \quad \Rightarrow \quad \sin^2\theta = \frac{1}{4}. \] Consequently, \( \sin\theta = \pm \frac{1}{2} \).2. These values are then substituted into the second equation, \( 2\cos^2\theta = 3\sin\theta \). Considering the solutions for \( \theta \) within the interval [0, \( 2\pi \)], the sum of these solutions is found to be \( \pi \).Therefore, the total sum of all solutions for \( \theta \) is \( \pi \).