Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>The sum of all values of \(\theta \in \left(0, \dfrac{\pi}{2}\right)\) satisfying \(\sin^2 2\theta + \cos^4 2\theta = \dfrac{3}{4}\) is:</p>
<p>\(\pi\)</p>
<p>\(\dfrac{5\pi}{4}\)</p>
<p>\(\dfrac{\pi}{2}\)</p>
<p>\(\dfrac{3\pi}{8}\)</p>
Step-by-Step Solution
Key Concept: Substitute u = sin²2θ to convert the trigonometric equation into a quadratic equation, then use sin²2θ + cos²2θ = 1 to eliminate the cosine term. The key is recognizing that cos⁴2θ = (1 - sin²2θ)² expands to a quadratic in sin²2θ.
<p><strong>Step 1:</strong> Let u = sin²2θ. Then cos²2θ = 1 - u, so cos⁴2θ = (1-u)²</p><p><strong>Step 2:</strong> Substitute into the equation: u + (1-u)² = 3/4</p><p>u + 1 - 2u + u² = 3/4</p><p>u² - u + 1/4 = 0</p><p>(u - 1/2)² = 0 ⟹ u = 1/2</p><p><strong>Step 3:</strong> Therefore sin²2θ = 1/2, giving sin2θ = ±1/√2</p><p><strong>Step 4:</strong> For θ ∈ (0, π/2), we have 2θ ∈ (0, π)</p><p>• sin2θ = 1/√2 ⟹ 2θ = π/4 or 3π/4 ⟹ θ = π/8 or 3π/8</p><p>• sin2θ = -1/√2 ⟹ 2θ ∉ (0, π) in this case</p><p><strong>Step 5:</strong> Both θ = π/8 and θ = 3π/8 lie in (0, π/2) and satisfy the original equation</p><p><strong>Step 6:</strong> Sum = π/8 + 3π/8 = 4π/8 = π/2</p><p>∴ Answer: C (π/2)</p>
Correct Answer: C