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TMUA 2021 · Paper 1 · Question 6 of 20

TMUA 2021 Paper 1 Question 6

Trigonometry — Trigonometric identities · a fraction that cancels. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1TrigonometryTrigonometric identities · a fraction that cancels6 options
The function f is given by f(x) = cos x + 37 + 5cos x − sin2 x Find the positive difference between the maximum and the minimum values of f(x).
  1. A0
  2. B13
  3. C12
  4. D23
  5. E1
  6. F2
Show the answer and worked solution
answer · D
  1. A0
  2. B13
  3. C12
  4. D23
  5. E1
  6. F2
Turn the denominator into a quadratic in cos x using sin2 x = 1 − cos2 x: it becomes cos2 x + 5cos x + 6 = (cos x + 2)(cos x + 3). The factor cos x + 3 is never zero, so it cancels and f(x) = 1cos x + 2. As cos x runs over [−1,  1], the denominator runs over [1,  3] and f runs over [13,  1]. The difference is 1 − 13 = 23.