Question

If \( 2 \sin \left( \frac{\pi}{3} - 2x \right) - 1 = 0 \), \( 0<x<\frac{\pi}{2} \), then the value of \( x \) is:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{5\pi}{12} \)
• D. \( \frac{\pi}{12} \)
• E. \( \frac{\pi}{6} \)

Hint:

To solve trigonometric equations, isolate the trigonometric function and use known angle values for sine or cosine.

Answer 1

The Correct Option is D

Step 1: Solve the given equation: \[ 2 \sin \left( \frac{\pi}{3} - 2x \right) = 1 \] \[ \sin \left( \frac{\pi}{3} - 2x \right) = \frac{1}{2} \] The solution to \( \sin \theta = \frac{1}{2} \) is \( \theta = \frac{\pi}{6} \). Thus: \[ \frac{\pi}{3} - 2x = \frac{\pi}{6} \] \[ 2x = \frac{\pi}{6} \] \[ x = \frac{\pi}{12} \]