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4PM1/02R Jan 2020, Question 8: Solving Trig Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 10 marks

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The question, in short

Use the double-angle formula to factorise a trigonometric expression g(A) into two brackets, then solve g(2θ) = 0 in radians.

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

Part (a) · 1 mark

The constant in the double-angle formula for sin 2A.

  1. 1sin 2A = 2 sin A cos A, so k = 2

Answerk = 2

Part (b) · 3 marks

Find the integers p, q and r that make the given factorised form equal to g(A).

  1. 12 − 2cos²A = 2 sin²A and 3 sin 2A = 6 sin A cos A
  2. 2g(A) = 3cos A − sin A − 6 sin A cos A + 2 sin²A
  3. 3= 3cos A(1 − 2 sin A) − sin A(1 − 2 sin A)
  4. 4= (3cos A − sin A)(1 − 2 sin A)
  5. 5p = 3, q = 1, r = 2

Answerp = 3, q = 1, r = 2

Part (c) · 6 marks

Solve g(2θ) = 0 for 0 ≤ θ < π.

  1. 1(3cos 2θ − sin 2θ)(1 − 2 sin 2θ) = 0, with 0 ≤ 2θ < 2π
  2. 2tan 2θ = 3: 2θ = 1.2490, 4.3906, so θ = 0.625, 2.20
  3. 3sin 2θ = 1/2: 2θ = π/6, 5π/6, so θ = π/12, 5π/12
  4. 4θ = 0.262, 1.31

Answerθ = 0.625, 2.20, π/12 (0.262), 5π/12 (1.31)

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