Question

Find the value of \[ \sin \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right). \] The answer is:

• A. \( \sqrt{3} \)
• B. \( \frac{\sqrt{3}}{2} \)
• C. \( \frac{\sqrt{3}}{4} \)
• D. \( -\sqrt{3} \)
• E. \( \frac{1}{\sqrt{3}} \)

Hint:

For problems involving inverse trigonometric functions and double angle formulas, remember to apply the basic trigonometric identities such as \( \sin^2 \theta + \cos^2 \theta = 1 \) to simplify the calculations.

Answer 1

The Correct Option is B

We are asked to evaluate: \[ \sin \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right). \] Let \( \theta = \sin^{-1} \left( \frac{1}{2} \right) \), so that \( \sin \theta = \frac{1}{2} \).
Step 1: Recall the double angle identity for sine: \[ \sin(2\theta) = 2 \sin \theta \cos \theta. \] Step 2: From \( \sin \theta = \frac{1}{2} \), we can find \( \cos \theta \) using the Pythagorean identity: \[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left( \frac{1}{2} \right)^2 = 1 - \frac{1}{4} = \frac{3}{4}. \] Thus, \[ \cos \theta = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}. \] Step 3: Now, apply the double angle formula: \[ \sin(2\theta) = 2 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{2}. \] Thus, the value of \( \sin \left( 2 \sin^{-1} \left( \frac{1}{2} \right) \right) \) is \( \frac{\sqrt{3}}{2} \).
Therefore, the correct answer is option (B).