Trigonometry & Inverse Trigonometry
Periodicity of trigonometric functions
Grade 11
Question:
<p>The period of the function \( f(x) = \sin^4 x + \cos^4 x \) is:</p>
<p>\( \pi \)</p>
<p>\( \dfrac{\pi}{2} \)</p>
<p>\( 2\pi \)</p>
<p>\( \dfrac{\pi}{4} \)</p>
Step-by-Step Solution
Key Concept: Simplify sin⁴x + cos⁴x using the algebraic identity (a² + b²)² = a⁴ + b⁴ + 2a²b², then recognize the resulting expression contains only cos(2x) terms, which determines the period.
<p><strong>Step 1:</strong> Use the identity (a² + b²)² = a⁴ + b⁴ + 2a²b²</p><p>sin⁴x + cos⁴x = (sin²x + cos²x)² - 2sin²x·cos²x</p><p>= 1 - 2sin²x·cos²x</p><p><strong>Step 2:</strong> Simplify using sin(2x) = 2sinx·cosx, so sin²x·cos²x = sin²(2x)/4</p><p>f(x) = 1 - 2·sin²(2x)/4 = 1 - sin²(2x)/2</p><p><strong>Step 3:</strong> Use sin²(2x) = (1 - cos(4x))/2</p><p>f(x) = 1 - (1 - cos(4x))/4 = 1 - 1/4 + cos(4x)/4 = 3/4 + cos(4x)/4</p><p><strong>Step 4:</strong> The function now contains only cos(4x). Since the period of cos(4x) is 2π/4 = π/2, and this is the only periodic term, the period of f(x) is <strong>π/2</strong>.</p><p>∴ Answer: B</p>
Correct Answer: B