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Find the number of solutions in [0, 2π) of the equation cos x + cos…
Question
Find the number of solutions in [0, 2π) of the equation cos x + cos 2x + cos 3x = 0.
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Step-by-step solution
Pair the outer terms: cos x + cos 3x = 2 cos 2x cos x. So the equation becomes 2 cos 2x cos x + cos 2x = cos 2x (2 cos x + 1) = 0.
Case 1: cos 2x = 0. In [0, 2π), 2x ∈ [0, 4π), and cos = 0 at 2x = π/2, 3π/2, 5π/2, 7π/2, giving x = π/4, 3π/4, 5π/4, 7π/4 — 4 solutions.
Case 2: 2 cos x + 1 = 0, i.e. cos x = −1/2. In [0, 2π) this gives x = 2π/3 and x = 4π/3 — 2 solutions.
No overlaps (cos x = −1/2 forces cos 2x = 2(1/4) − 1 = −1/2 ≠ 0). Total = 4 + 2 = 6.
Final answer6
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