Trigonometry & Inverse Trigonometry
Modulus of Trigonometric Functions
Grade 11
Question:
<p>If \(|\sin x + \cos x| = |\sin x| + |\cos x|\) (\(\sin x, \cos x \neq 0\)), then in which quadrant does \(x\) lie?</p>
<p>First or second quadrant</p>
<p>Second or third quadrant</p>
<p>First or third quadrant</p>
<p>Third or fourth quadrant</p>
Step-by-Step Solution
Key Concept: The equation |sin x + cos x| = |sin x| + |cos x| holds only when sin x and cos x have the same sign, because the absolute value of a sum equals the sum of absolute values only when all terms share the same sign.
<p><strong>Step 1:</strong> Recall that |a + b| = |a| + |b| if and only if a and b have the same sign (both non-negative or both non-positive).</p><p><strong>Step 2:</strong> Apply this to |sin x + cos x| = |sin x| + |cos x|. This equality holds when sin x and cos x have the same sign.</p><p><strong>Step 3:</strong> Analyze by quadrant:</p><ul><li><strong>Quadrant I:</strong> sin x > 0, cos x > 0 ✓ (same sign)</li><li><strong>Quadrant II:</strong> sin x > 0, cos x < 0 ✗ (opposite signs)</li><li><strong>Quadrant III:</strong> sin x < 0, cos x < 0 ✓ (same sign)</li><li><strong>Quadrant IV:</strong> sin x < 0, cos x > 0 ✗ (opposite signs)</li></ul><p><strong>Step 4:</strong> The condition is satisfied in Quadrants I and III.</p><p>∴ Answer: C (Quadrants I and III)</p>
Correct Answer: C