<p>Given \(\sin^2 2\theta + \cos^4 2\theta = \dfrac{3}{4}\), find the sum of all values of \(\theta \in \left[0, \dfrac{\pi}{2}\right]\).</p>
Step-by-Step Solution
Key Concept: Convert the equation using sin²2θ + cos²2θ = 1 to get a quadratic in cos²2θ, then use the constraint that cos²2θ ∈ [0,1] to find valid solutions for 2θ, and finally sum all θ values in the given interval.
<p><strong>Step 1:</strong> Let sin²2θ = 1 - cos²2θ. Substitute into the equation:</p><p>(1 - cos²2θ) + cos⁴2θ = 3/4</p><p>cos⁴2θ - cos²2θ + 1/4 = 0</p><p><strong>Step 2:</strong> Let x = cos²2θ. Then: x² - x + 1/4 = 0, which factors as (x - 1/2)² = 0</p><p>So cos²2θ = 1/2, giving cos2θ = ±1/√2</p><p><strong>Step 3:</strong> For θ ∈ [0, π/2], we have 2θ ∈ [0, π].</p><p>When cos2θ = 1/√2: 2θ = π/4, so θ = π/8</p><p>When cos2θ = -1/√2: 2θ = 3π/4, so θ = 3π/8</p><p><strong>Step 4:</strong> Both values lie in [0, π/2]. Sum = π/8 + 3π/8 = 4π/8</p><p>∴ Answer: <strong>π/2</strong></p>
Correct Answer: π/2