Question

If \( \sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = 1 \), then the value of \( x \) in the interval \( [0, \pi] \) is:

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( 0 \)
• D. \( \frac{\pi}{4} \)

Hint:

Sum of Sine Functions}
Use sum and difference identities
Simplify using known values like \( \cos \frac{\pi}{3} = \frac{1}{2} \)
Solve resulting equation within given interval

Answer 1

The Correct Option is A

Use identity: \[ \sin(a + b) + \sin(a - b) = 2 \sin a \cos b \] So: \[ \sin\left(x + \frac{\pi}{3}\right) + \sin\left(x - \frac{\pi}{3}\right) = 2 \sin x \cos \frac{\pi}{3} = 2 \sin x \cdot \frac{1}{2} = \sin x \] Given: \[ \sin x = 1 \Rightarrow x = \frac{\pi}{2} \]