Question

If \(\sin \theta = \frac{1}{2}\) and \(\theta\) is acute, then the value of \(\text{sin}\ 2\theta\) is 

• A. 1
• B. \(\frac{\sqrt{3}}{2}\)
• C. \(\frac{1}{2}\)
• D. \(-\frac{\sqrt{3}}{2}\)

Answer 1

The Correct Option is B

Explanation:
Given: \[ \sin \theta = \frac{1}{2} \] and \( \theta \) is acute. This implies that \( \theta = 30^\circ \), since \( \sin 30^\circ = \frac{1}{2} \). Now, using the double angle identity for sine: \[ \sin 2\theta = 2 \sin \theta \cos \theta \] Substituting \( \sin \theta = \frac{1}{2} \) and \( \cos 30^\circ = \frac{\sqrt{3}}{2} \): \[ \sin 2\theta = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2} \]

Hence, the value of \( \sin 2\theta \) is \(\frac{\sqrt{3}}{2}\).