Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>Given \(f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)\), find the value of \(f_4(x) - f_6(x)\).</p>
Step-by-Step Solution
Key Concept: Express sin⁴x + cos⁴x and sin⁶x + cos⁶x in terms of sin²x·cos²x using algebraic identities, then compute their difference systematically.
<p><strong>Step 1:</strong> Find f₄(x) = ¼(sin⁴x + cos⁴x)</p><p>Use: sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x·cos²x = 1 - 2sin²x·cos²x</p><p>Therefore: f₄(x) = ¼(1 - 2sin²x·cos²x)</p><p><strong>Step 2:</strong> Find f₆(x) = ⅙(sin⁶x + cos⁶x)</p><p>Use: sin⁶x + cos⁶x = (sin²x + cos²x)³ - 3sin²x·cos²x(sin²x + cos²x) = 1 - 3sin²x·cos²x</p><p>Therefore: f₆(x) = ⅙(1 - 3sin²x·cos²x)</p><p><strong>Step 3:</strong> Compute f₄(x) - f₆(x)</p><p>f₄(x) - f₆(x) = ¼(1 - 2sin²x·cos²x) - ⅙(1 - 3sin²x·cos²x)</p><p>= ¼ - ½sin²x·cos²x - ⅙ + ½sin²x·cos²x</p><p>= ¼ - ⅙ = (3 - 2)/12 = 1/12</p><p>∴ Answer: 0.0833 (or 1/12)</p>
Correct Answer: 0.0833