<p>\(\sin^{-1}(3x - 4x^3) = \lambda \sin^{-1} x\) then \(\lambda = \underline{\quad}\).</p>
Step-by-Step Solution
Key Concept: Recognize that 3x - 4x³ is the sine triple angle formula: sin(3θ) = 3sinθ - 4sin³θ. If we set sinθ = x, then sin(3θ) = 3x - 4x³, which means sin⁻¹(3x - 4x³) = 3sin⁻¹(x) in the valid domain.
<p><strong>Step 1:</strong> Recognize the structure of 3x - 4x³. This matches the sine triple angle formula: sin(3θ) = 3sinθ - 4sin³θ.</p><p><strong>Step 2:</strong> Let sin⁻¹(x) = θ, so x = sinθ where θ ∈ [-π/6, π/6] (restricted domain).</p><p><strong>Step 3:</strong> Then 3x - 4x³ = 3sinθ - 4sin³θ = sin(3θ).</p><p><strong>Step 4:</strong> Therefore, sin⁻¹(3x - 4x³) = sin⁻¹(sin(3θ)) = 3θ = 3sin⁻¹(x).</p><p><strong>Step 5:</strong> Comparing with sin⁻¹(3x - 4x³) = λsin⁻¹(x), we get λ = 3.</p><p>∴ Answer: <strong>3</strong></p>
Correct Answer: 3