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TMUA 2023 · Paper 1 · Question 12 of 20

TMUA 2023 Paper 1 Question 12

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

TMUA 2023 · Paper 1TrigonometryTrigonometric equations with indices7 options
How many solutions are there to the equation 2tan2 x4sin2 x = 1 in the range 0 ≤ x ≤ 2π?
  1. A2
  2. B3
  3. C4
  4. D5
  5. E6
  6. F7
  7. G8
Show the answer and worked solution
answer · F
  1. A2
  2. B3
  3. C4
  4. D5
  5. E6
  6. F7
  7. G8
Writing 4sin2 x = 22sin2 x, the equation becomes 2tan2 x − 2sin2 x = 1, so tan2 x = 2sin2 x. Then sin2 xcos2 x − 2sin2 x = 0, that is sin2 x(1cos2 x − 2) = 0. So sin x = 0, giving x = 0,  π,  2π, or cos2 x = 12, giving x = π4,  3π4,  5π4,  7π4. None makes tan x undefined, so there are 7 solutions.