Trigonometry & Inverse Trigonometry
Periodic Functions
Grade 11
Question:
<p>Let period of \(f(x) = \frac{|\sin x| - |\cos x|}{|\sin x + \cos x|}\) is <span style='font-style:italic;'>l</span>, then [4<span style='font-style:italic;'>l</span>] is equal to __________ where []× denotes greatest integer function.</p>
Step-by-Step Solution
Key Concept: To find the period of f(x), we must simplify the function using properties of absolute values and periodic functions. The period is the smallest positive value T such that f(x+T) = f(x) for all x in the domain.
<p><strong>Step 1: Analyze the domain and simplify.</strong></p><p>The function f(x) = (|sin x| - |cos x|)/(|sin x + cos x|) is defined when sin x + cos x ≠ 0.</p><p><strong>Step 2: Determine periodicity of numerator and denominator separately.</strong></p><p>• |sin x| has period π</p><p>• |cos x| has period π</p><p>• Therefore, |sin x| - |cos x| has period π</p><p>• |sin x + cos x| = |√2 sin(x + π/4)| has period π</p><p><strong>Step 3: Check if period of f(x) is π.</strong></p><p>Verify: f(x + π) = (|sin(x+π)| - |cos(x+π)|)/(|sin(x+π) + cos(x+π)|)</p><p>= (|-sin x| - |-cos x|)/(|-sin x - cos x|)</p><p>= (|sin x| - |cos x|)/(|sin x + cos x|) = f(x) ✓</p><p><strong>Step 4: Verify π is the fundamental period.</strong></p><p>Check that π/2 is NOT a period:</p><p>f(x + π/2) = (|sin(x+π/2)| - |cos(x+π/2)|)/(|sin(x+π/2) + cos(x+π/2)|)</p><p>= (|cos x| - |-sin x|)/(|cos x - sin x|)</p><p>= (|cos x| - |sin x|)/(|cos x - sin x|) ≠ f(x) in general</p><p><strong>Step 5: Calculate [4l].</strong></p><p>The period is l = π.</p><p>Therefore, 4l = 4π ≈ 4 × 3.14159... = 12.566...</p><p>[4l] = [12.566...] = 12</p><p><strong>∴ Answer: 12</strong></p>
Correct Answer: 12