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TMUA 2020 · Paper 1 · Question 18 of 20

TMUA 2020 Paper 1 Question 18

Trigonometry — Trigonometric equation with a modulus. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1TrigonometryTrigonometric equation with a modulus6 options
Find the number of solutions and the sum of the solutions of the equation 1 − 2cos2 x = |cos x| where 0 ≤ x ≤ 180∘.
  1. ANumber of solutions = 2, sum of solutions = 180∘
  2. BNumber of solutions = 2, sum of solutions = 240∘
  3. CNumber of solutions = 3, sum of solutions = 180∘
  4. DNumber of solutions = 3, sum of solutions = 360∘
  5. ENumber of solutions = 4, sum of solutions = 240∘
  6. FNumber of solutions = 4, sum of solutions = 360∘
Show the answer and worked solution
answer · A
  1. ANumber of solutions = 2, sum of solutions = 180∘
  2. BNumber of solutions = 2, sum of solutions = 240∘
  3. CNumber of solutions = 3, sum of solutions = 180∘
  4. DNumber of solutions = 3, sum of solutions = 360∘
  5. ENumber of solutions = 4, sum of solutions = 240∘
  6. FNumber of solutions = 4, sum of solutions = 360∘
Set u = |cos x|, so u ≥ 0 and cos2 x = u2. The equation becomes 1 − 2u2 = u, that is 2u2 + u − 1 = 0, which factorises as (2u − 1)(u + 1) = 0. The root u = −1 is impossible for a modulus, leaving |cos x| = 12, so cos x = 12 or cos x = −12. In 0 ≤ x ≤ 180∘ that gives x = 60∘ and x = 120∘: two solutions summing to 180∘.