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TMUA 2021 · Paper 1 · Question 19 of 20

TMUA 2021 Paper 1 Question 19

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

TMUA 2021 · Paper 1TrigonometryTrigonometric equations · counting solutions to fix a range6 optionshard
The equation sin2(4cosθ × 60∘) = 34 has exactly three solutions in the range 0∘ ≤ θ ≤ x∘

What is the range of all possible values of x?

  1. A90 ≤ x < 120
  2. B90 ≤ x < 270
  3. C120 ≤ x < 240
  4. D270 ≤ x < 300
  5. E300 ≤ x < 360
  6. F450 ≤ x < 630
Show the answer and worked solution
answer · B
  1. A90 ≤ x < 120
  2. B90 ≤ x < 270
  3. C120 ≤ x < 240
  4. D270 ≤ x < 300
  5. E300 ≤ x < 360
  6. F450 ≤ x < 630
Write t = 4cosθ, so the angle is 60t degrees and the equation is sin(60t) = ±√32, which holds exactly when 60t degrees differs from 60∘ or from 120∘ by a whole number of 180∘ — that is, when t is an integer not divisible by 3. Since cosθ ∈ [−1, 1], t lies in [14,  4], so the usable values are t = 1,  2,  4, giving cosθ = 0,  12,  1. Listing the resulting θ in order from 0∘: 0∘,  60∘,  90∘,  270∘,  300∘,  360∘. Exactly three of them lie in [0∘,  x∘] when x has reached 90 but not yet 270, so 90 ≤ x < 270.