Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If <span>|sin x + cos x| = |sin x| + |cos x|</span>, <span>x ∈ [0, 2π]</span> then the solution set is</p>
<p>(a) <span>[0, π/2]</span></p>
<p>(b) <span>[0, π/2] ∪ [π, 3π/2] ∪ {2π}</span></p>
<p>(c) <span>[π, 3π/2] ∪ {2π}</span></p>
<p>(d) <span>[0, 2π]</span></p>
Step-by-Step Solution
Key Concept: The absolute value equation holds when the arguments inside have the same sign or are zero, which occurs in quadrants I and III.
<p><strong>Analysis:</strong> The equation <span>|sin x + cos x| = |sin x| + |cos x|</span> holds when <span>sin x</span> and <span>cos x</span> have the same sign or when at least one is zero.</p><p>This occurs in quadrant I (where both are positive): <span>[0, π/2]</span></p><p>In quadrant III (where both are negative): <span>[π, 3π/2]</span></p><p>At the boundary point <span>x = 2π</span> where <span>sin(2π) = 0, cos(2π) = 1</span>: the condition holds.</p><p>∴ Solution set is <span>[0, π/2] ∪ [π, 3π/2] ∪ {2π}</span></p>
Correct Answer: b