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Find the number of values of x in the interval [0, 2π) that satisfy…
Question
Find the number of values of x in the interval [0, 2π) that satisfy the equation sin(5x) = sin(3x) + sin(x).
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Step-by-step solution
Group the equation as sin(5x) − sin(3x) = sin(x).
Using sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2): sin(5x) − sin(3x) = 2 cos(4x) sin(x).
So the equation becomes 2 cos(4x) sin(x) = sin(x), i.e.
sin(x)(2 cos(4x) − 1) = 0.
Case 1: sin(x) = 0. In [0, 2π) this gives x = 0 and x = π — 2 solutions.
Case 2: cos(4x) = 1/2.
As x runs over [0, 2π), 4x runs over [0, 8π).
cos θ = 1/2 occurs twice per 2π period, and [0, 8π) spans 4 full periods, giving 4 × 2 = 8 solutions.
None of these coincide with x = 0 or π (there cos(4x) = 1 ≠ 1/2), so there is no overlap.
Total = 2 + 8 = 10.
Final answer10
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