36 exam-style questions on Solving trigonometric equations — topic 3.8 of Geometry and Trigonometry in the IB Mathematics: Analysis and Approaches syllabus — with answers and explanations. Samples below; the full set is free with an account.
Solve sin x = 1/2 for 0 ≤ x ≤ 2π.
Answer: x = π/6 and x = 5π/6
sin is positive in the first and second quadrants: x = π/6 and x = π − π/6 = 5π/6.
Solve 2 cos x + 1 = 0 for 0 ≤ x < 2π.
Answer: x = 2π/3 and x = 4π/3
cos x = −1/2, and cosine is negative in the second and third quadrants: x = π − π/3 = 2π/3 and x = π + π/3 = 4π/3.
Determine all solutions of tan x = √3 in 0 ≤ x < 2π.
Answer: x = π/3 and x = 4π/3
tan has period π, so the second solution is π/3 + π = 4π/3; tangent is positive in the first and third quadrants.
How many solutions does the equation sin(2x) = 1/2 have in 0 ≤ x < 2π?
Answer: 4
Let u = 2x, so u runs over [0, 4π) — two full sine cycles, each giving 2 solutions, so 4 in total.
How many solutions does 2 sin²x − sin x − 1 = 0 have in 0 ≤ x < 2π?
Answer: 3
Factorising gives (2 sin x + 1)(sin x − 1) = 0, so sin x = −1/2 (x = 7π/6, 11π/6) or sin x = 1 (x = π/2): three solutions.
Solve cos(2x) = 1 for 0 ≤ x < 2π.
Answer: x = 0 and x = π
cos(2x) = 1 requires 2x = 0, 2π, 4π, …; with 2x in [0, 4π) that gives 2x = 0 and 2x = 2π, so x = 0 and x = π.
That was 6 of 36 Solving trigonometric equations questions.
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