<p>If \( \sin^{-1}\dfrac{x}{5} + \sin^{-1}\dfrac{4}{5} = \dfrac{\pi}{2} \), then \( x \) equals:</p>
Step-by-Step Solution
Key Concept: Use the complementary property: sin⁻¹(a) + sin⁻¹(b) = π/2 implies sin⁻¹(a) = cos⁻¹(b), which means a = √(1-b²). Alternatively, recognize that sin⁻¹(a) + cos⁻¹(a) = π/2, so sin⁻¹(4/5) = cos⁻¹(4/5) tells us the complementary angle relationship.
<p><strong>Step 1:</strong> Given: sin⁻¹(x/5) + sin⁻¹(4/5) = π/2</p><p><strong>Step 2:</strong> Rearrange: sin⁻¹(x/5) = π/2 - sin⁻¹(4/5) = cos⁻¹(4/5)</p><p><strong>Step 3:</strong> Apply the identity cos⁻¹(a) = sin⁻¹(√(1-a²)). Therefore: sin⁻¹(x/5) = sin⁻¹(√(1-(4/5)²))</p><p><strong>Step 4:</strong> Simplify: √(1 - 16/25) = √(9/25) = 3/5</p><p><strong>Step 5:</strong> Thus: x/5 = 3/5, so <strong>x = 3</strong></p><p><strong>Verification:</strong> sin⁻¹(3/5) + sin⁻¹(4/5) = π/2 ✓ (since 3-4-5 is a Pythagorean triple)</p><p>∴ Answer: B</p>
Correct Answer: B