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TMUA 2023 · Paper 1 · Question 9 of 20

TMUA 2023 Paper 1 Question 9

Trigonometry — Trigonometric equations · counting solutions. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1TrigonometryTrigonometric equations · counting solutions6 options
How many solutions are there to (1 + 3cos 3θ)2 = 4 in the interval 0∘ ≤ θ ≤ 180∘?
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
  6. F6
Show the answer and worked solution
answer · E
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
  6. F6
Taking square roots, 1 + 3cos 3θ = ± 2, so cos 3θ = 13 or cos 3θ = −1. As θ runs over [0∘,  180∘], 3θ runs over [0∘,  540∘]. In that range cos u = −1 at u = 180∘ and 540∘ — two solutions — while cos u = 13 has one solution in each of [0∘, 180∘], [180∘, 360∘] and [360∘, 540∘] — three more. That is 5 in total.