Trigonometry & Inverse Trigonometry
Trigonometric Identities
Grade 11
Question:
<p>Let \(f_k(x) = \dfrac{1}{k}(\sin^k x + \cos^k x)\) for \(k = 1, 2, 3, \ldots\) Then for all \(x \in R\), the value of \(f_4(x) - f_6(x)\) is equal to __________ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Express f₄(x) - f₆(x) by factoring out common terms and using the identity sin²x + cos²x = 1 to reduce the expression to a constant independent of x.
<p><strong>Step 1:</strong> Write out the expressions:</p><p>f₄(x) = ¼(sin⁴x + cos⁴x)</p><p>f₆(x) = ⅙(sin⁶x + cos⁶x)</p><p><strong>Step 2:</strong> Express sin⁴x + cos⁴x using sin²x + cos²x = 1:</p><p>sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x cos²x = 1 - 2sin²x cos²x</p><p><strong>Step 3:</strong> Express sin⁶x + cos⁶x using the identity:</p><p>sin⁶x + cos⁶x = (sin²x + cos²x)³ - 3sin²x cos²x(sin²x + cos²x) = 1 - 3sin²x cos²x</p><p><strong>Step 4:</strong> Calculate 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 = (6 - 4)/24 = 2/24 = 1/12</p><p><strong>Step 5:</strong> Convert to decimal: 1/12 = 0.08333...</p><p>∴ Answer: <strong>0.0833</strong></p>
Correct Answer: 0.0833