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WMA12/01 Jun 2023, Question 9: Solving Trig Equations

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

Prove a trigonometric identity reduces to a quadratic in cos θ, then use it to find all angles x in 0 < x < 360° satisfying the same relationship with 2x, to one decimal place.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Starting from 3 cos θ (tan θ sin θ + 3) = 11 − 5 cos θ, show by algebraic manipulation that it simplifies to 3 cos²θ − 14 cos θ + 8 = 0.

  1. 1tan θ = sin θ / cos θ, so 3 cos θ × (sin θ/cos θ) × sin θ = 3 sin²θ
  2. 23 sin²θ + 9 cos θ = 11 − 5 cos θ
  3. 33(1 − cos²θ) + 9 cos θ = 11 − 5 cos θ
  4. 43 − 3 cos²θ + 14 cos θ − 11 = 0
  5. 53 cos²θ − 14 cos θ + 8 = 0

AnswerShown: 3 sin²θ + 9 cos θ = 11 − 5 cos θ; 3(1 − cos²θ) + 14 cos θ − 11 = 0 gives 3 cos²θ − 14 cos θ + 8 = 0

Part (b) · 4 marks

Using the result from part a with 2x substituted for θ, determine every x in 0 < x < 360° that satisfies the equation, rounded to one decimal place.

  1. 1With θ = 2x: 3 cos²2x − 14 cos 2x + 8 = 0
  2. 2(3 cos 2x − 2)(cos 2x − 4) = 0
  3. 3cos 2x = 4 has no solutions, so cos 2x = 2/3
  4. 42x = 48.19°, 311.81°, 408.19°, 671.81° (for 0 < 2x < 720°)
  5. 5x = 24.1°, 155.9°, 204.1°, 335.9°

Answerx = 24.1°, 155.9°, 204.1°, 335.9°

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