Kerala KEAM 5 June 2024 Solved Paper

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).Quick Tip: 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.