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TMUA 2022 · Paper 1 · Question 1 of 20

TMUA 2022 Paper 1 Question 1

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

TMUA 2022 · Paper 1TrigonometryTrigonometric equations · hidden quadratic8 options
How many real solutions are there to the equation 2cos4θ − 5cos2θ + 3 = 0 in the interval 0 ≤ θ ≤ 2π?
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
  6. F6
  7. G7
  8. H8
Show the answer and worked solution
answer · C
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
  6. F6
  7. G7
  8. H8
Put u = cos2θ, so 2u2 − 5u + 3 = 0 and (2u−3)(u−1) = 0. The root u = 32 is impossible because cos2θ ≤ 1, leaving cos2θ = 1, that is cosθ = ± 1. In [0,  2π] that gives θ = 0, π and 2π — three solutions.