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TMUA 2022 · Paper 1 · Question 9 of 20

TMUA 2022 Paper 1 Question 9

Trigonometry — Simultaneous functional equations. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1TrigonometrySimultaneous functional equations7 optionshard
This question is about pairs of functions f and g that satisfy f(x) − g(x) = 2sin x f(x) g(x) = cos2 x for all real numbers x.

Across all solutions for f(x), what is the minimum value that f(x) attains for any x?

  1. A1 − √2
  2. B−1 − √2
  3. C0
  4. D−1
  5. E−2
  6. F−3
  7. G−4
Show the answer and worked solution
answer · E
  1. A1 − √2
  2. B−1 − √2
  3. C0
  4. D−1
  5. E−2
  6. F−3
  7. G−4
Substituting g = f − 2sin x into the product gives f2 − 2fsin x − cos2 x = 0. Solving as a quadratic in f, f = sin x ± √sin2 x + cos2 x = sin x ± 1. The branch f = sin x − 1 dips lowest, reaching −2 when sin x = −1.