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TMUA 2016 · Paper 2 · Question 3 of 20

TMUA 2016 Paper 2 Question 3

Trigonometry — Using sin2 x + cos2 x = 1 · all solutions in a range. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2TrigonometryUsing sin2 x + cos2 x = 1 · all solutions in a range6 options
What is the value, in radians, of the largest angle x in the range 0 ≤ x ≤ 2π that satisfies the equation 8sin2 x + 4cos2 x = 7?
  1. A2π3
  2. B5π6
  3. C4π3
  4. D5π3
  5. E7π4
  6. F11π6
Show the answer and worked solution
answer · D
  1. A2π3
  2. B5π6
  3. C4π3
  4. D5π3
  5. E7π4
  6. F11π6
Write 4cos2 x = 4 − 4sin2 x, so the left-hand side collapses to 4sin2 x + 4. Then sin2 x = 34, giving sin x = ±√32. In 0 ≤ x ≤ 2π that is x = π3,  2π3,  4π3,  5π3. The largest is 5π3. Losing the negative square root is what makes people stop at 2π3.