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WMA12/01 Oct 2022, Question 5: Solving Trig Equations

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Show detailed reasoning throughout, with no reliance on calculator-only methods: transform a trigonometric equation into a quadratic in sin θ, then solve the related equation in 2x over −π/2 ≤ x ≤ π/2.

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

Part (a) · 3 marks

Show that (3 cos θ − tan θ) cos θ = 2 rearranges to 3 sin²θ + sin θ − 1 = 0, giving full algebraic reasoning.

  1. 1tan θ = sin θ / cos θ, so (3cos θ − sin θ/cos θ) cos θ = 2
  2. 23cos²θ − sin θ = 2
  3. 3Use cos²θ = 1 − sin²θ: 3(1 − sin²θ) − sin θ = 2
  4. 43sin²θ + sin θ − 1 = 0

AnswerShown: 3cos²θ − sin θ = 2 ⇒ 3(1 − sin²θ) − sin θ = 2 ⇒ 3sin²θ + sin θ − 1 = 0

Part (b) · 5 marks

Hence solve (3 cos 2x − tan 2x) cos 2x = 2 for −π/2 ≤ x ≤ π/2, using part a and giving all solutions.

  1. 1With θ = 2x: 3sin²2x + sin 2x − 1 = 0
  2. 2sin 2x = (−1 ± √13)/6, so sin 2x = 0.4343… or −0.7676…
  3. 3−π ≤ 2x ≤ π
  4. 4sin 2x = 0.4343…: 2x = 0.4492 or π − 0.4492 = 2.6924, so x = 0.225, 1.35
  5. 5sin 2x = −0.7676…: 2x = −0.8751 or −π + 0.8751 = −2.2665, so x = −0.438, −1.13
  6. 6x = −1.13, −0.438, 0.225, 1.35

Answerx = −1.13, −0.438, 0.225, 1.35

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