Trigonometry & Inverse Trigonometry
Multiple angle formulas
Grade 11
Question:
<p>If \(a \cos^2 3a + b \cos^4 a = 16 \cos^6 a + 9 \cos^2 a\) is an identity, then</p>
<p>(a) \(a = 1, b = 24\)</p>
<p>(b) \(a = 3, b = 24\)</p>
<p>(c) \(a = 4, b = 2\)</p>
<p>(d) \(a = 7, b = 18\)</p>
Step-by-Step Solution
Key Concept: Since the given equation is an identity (true for all values of a), we can substitute specific values of cos(a) to create a system of equations. By choosing convenient values like cos(a) = 0 and cos(a) = 1, we can determine the coefficients a and b uniquely.
<p><strong>Step 1: Set up the identity equation</strong></p><p>Given: a cos²(3a) + b cos⁴(a) = 16 cos⁶(a) + 9 cos²(a) is an identity.</p><p>Let x = cos(a) where 0 ≤ x ≤ 1. We need to express cos²(3a) in terms of cos(a).</p><p><strong>Step 2: Express cos(3a) in terms of cos(a)</strong></p><p>Using the triple angle formula: cos(3a) = 4cos³(a) - 3cos(a)</p><p>Therefore: cos²(3a) = (4cos³(a) - 3cos(a))² = (4x³ - 3x)²</p><p><strong>Step 3: Expand cos²(3a)</strong></p><p>cos²(3a) = 16x⁶ - 24x⁴ + 9x²</p><p>The identity becomes:</p><p>a(16x⁶ - 24x⁴ + 9x²) + bx⁴ = 16x⁶ + 9x²</p><p><strong>Step 4: Expand the left side</strong></p><p>16ax⁶ - 24ax⁴ + 9ax² + bx⁴ = 16x⁶ + 9x²</p><p>16ax⁶ + (b - 24a)x⁴ + 9ax² = 16x⁶ + 0·x⁴ + 9x²</p><p><strong>Step 5: Compare coefficients</strong></p><p>For this to be an identity, coefficients of like powers must be equal:</p><p>• Coefficient of x⁶: 16a = 16 ⟹ a = 1</p><p>• Coefficient of x⁴: b - 24a = 0 ⟹ b = 24a = 24(1) = 24</p><p>• Coefficient of x²: 9a = 9 ✓ (confirms a = 1)</p><p><strong>Step 6: Verify</strong></p><p>With a = 1 and b = 24: cos²(3a) + 24cos⁴(a) = 16cos⁶(a) + 24cos⁴(a) + 9cos²(a) - 24cos⁴(a) = 16cos⁶(a) + 9cos²(a) ✓</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A