Question

If \( \cos A = \frac{\sqrt{3}}{2} \), then find the value of \( \sin 2A \).

Hint:

To find \( \sin 2A \), use the double angle identity \( \sin 2A = 2 \sin A \cos A \), and apply the Pythagorean identity to find \( \sin A \).

Answer 1

We are given that \( \cos A = \frac{\sqrt{3}}{2} \). First, recall the double angle identity for sine: \[ \sin 2A = 2 \sin A \cos A. \] We already know that \( \cos A = \frac{\sqrt{3}}{2} \). To find \( \sin A \), use the Pythagorean identity: \[ \sin^2 A + \cos^2 A = 1. \] Substitute \( \cos A = \frac{\sqrt{3}}{2} \) into the equation: \[ \sin^2 A + \left( \frac{\sqrt{3}}{2} \right)^2 = 1, \] \[ \sin^2 A + \frac{3}{4} = 1. \] Now, subtract \( \frac{3}{4} \) from both sides: \[ \sin^2 A = 1 - \frac{3}{4} = \frac{1}{4}. \] Take the square root of both sides: \[ \sin A = \frac{1}{2} \quad \text{(since \( A \) is in the first quadrant, \( \sin A \) is positive)}. \] Now, substitute the values of \( \sin A \) and \( \cos A \) into the double angle identity: \[ \sin 2A = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}. \]
Conclusion:
Therefore, the value of \( \sin 2A \) is \( \frac{\sqrt{3}}{2} \).