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TMUA 2019 · Paper 1 · Question 14 of 20

TMUA 2019 Paper 1 Question 14

Trigonometry — Simultaneous trigonometric equations · solving for sin and cos. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 1TrigonometrySimultaneous trigonometric equations · solving for sin and cos6 options
x satisfies the simultaneous equations sin 2x + √3cos 2x = −1 and √3sin 2x − cos 2x = √3 where 0∘ ≤ x ≤ 360∘.

Find the sum of the possible values of x.

  1. A210∘
  2. B330∘
  3. C390∘
  4. D660∘
  5. E780∘
  6. F930∘
Show the answer and worked solution
answer · B
  1. A210∘
  2. B330∘
  3. C390∘
  4. D660∘
  5. E780∘
  6. F930∘
Treat s = sin 2x and c = cos 2x as two unknowns in a pair of linear equations. From the second, c = √3 s − √3; substituting into the first gives s + 3s − 3 = −1, so s = 12 and then c = −√32. A sine of 12 with a negative cosine puts 2x in the second quadrant: 2x = 150∘. Since 0∘ ≤ x ≤ 360∘ means 0∘ ≤ 2x ≤ 720∘, also 2x = 510∘. So x = 75∘ or 255∘, summing to 330∘.