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TMUA 2019 · Paper 1 · Question 17 of 20

TMUA 2019 Paper 1 Question 17

Trigonometry — Product of trigonometric factors · sign analysis on an interval. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1TrigonometryProduct of trigonometric factors · sign analysis on an interval8 optionshard
Find the fraction of the interval 0 ≤ θ ≤ π for which the inequality (sin 2θ − 12)(sinθ − cosθ) ≥ 0 is satisfied.
  1. A112
  2. B16
  3. C14
  4. D512
  5. E712
  6. F34
  7. G56
  8. H1112
Show the answer and worked solution
answer · C
  1. A112
  2. B16
  3. C14
  4. D512
  5. E712
  6. F34
  7. G56
  8. H1112
Find where each factor is non-negative, then take the two cases where the signs agree. Working in degrees, sin 2θ ≥ 12 needs 2θ ∈ [30∘,  150∘], i.e. θ ∈ [15∘,  75∘] (the next band, θ ∈ [195∘,  255∘], is outside the interval). For the second factor, sinθ − cosθ = √2sin(θ − 45∘) ≥ 0 needs θ ≥ 45∘. Both non-negative on [45∘,  75∘], length 30∘; both non-positive on [0∘,  15∘], length 15∘ (beyond 75∘ the second factor is positive while the first is negative). That is 45∘ out of 180∘, a fraction of 14.