<p>The number of solutions of the equation \(1 + \sin^4 x - \cos^2 3x\), \(x \in \left[-\dfrac{5\pi}{2}, \dfrac{5\pi}{2}\right]\) is __________ .</p>
Step-by-Step Solution
Key Concept: The equation 1 + sin⁴x = cos²3x requires recognizing that sin⁴x ≥ 0 makes LHS ≥ 1, while cos²3x ≤ 1, forcing both sides equal to 1. This means sin⁴x = 0 AND cos²3x = 1 simultaneously.
<p><strong>Step 1:</strong> Rewrite the equation: 1 + sin⁴x = cos²3x</p><p><strong>Step 2:</strong> Analyze ranges: Since sin⁴x ≥ 0, we have LHS = 1 + sin⁴x ≥ 1. Since cos²3x ≤ 1, we have RHS ≤ 1.</p><p><strong>Step 3:</strong> For equality to hold with LHS ≥ 1 and RHS ≤ 1, we need LHS = RHS = 1 exactly. This requires:</p><p>• sin⁴x = 0 ⟹ sin x = 0 ⟹ x = nπ, n ∈ ℤ</p><p>• cos²3x = 1 ⟹ cos 3x = ±1 ⟹ 3x = mπ ⟹ x = mπ/3, m ∈ ℤ</p><p><strong>Step 4:</strong> Find common solutions: x must satisfy both x = nπ AND x = mπ/3. This requires nπ = mπ/3, so m = 3n. Thus x = nπ where n ∈ ℤ.</p><p><strong>Step 5:</strong> Count solutions in [-5π/2, 5π/2]:</p><p>x = nπ where -5π/2 ≤ nπ ≤ 5π/2</p><p>-2.5 ≤ n ≤ 2.5</p><p>n ∈ {-2, -1, 0, 1, 2}</p><p>Solutions: x ∈ {-2π, -π, 0, π, 2π}</p><p><strong>∴ Answer: 5</strong></p>
Correct Answer: 5