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4PM1/01R Jan 2023, Question 10: Solving Trig Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 16 marks

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The question, in short

Prove a product identity from the compound-angle formulae, use it to solve a trig equation in terms of π, then evaluate a definite integral.

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

Part (a) · 2 marks

Use the compound-angle formulae to prove the product identity.

  1. 1cos(A − B) = cos A cos B + sin A sin B
  2. 2cos(A + B) = cos A cos B − sin A sin B
  3. 3Subtract: cos(A − B) − cos(A + B) = 2 sin A sin B

Answercos(A − B) − cos(A + B) = 2 sin A sin B shown

Part (b) · 1 mark

Apply it with 7θ and 2θ.

  1. 1A − B = 5θ and A + B = 9θ
  2. 2A = 7θ, B = 2θ
  3. 3cos 5θ − cos 9θ = 2 sin 7θ sin 2θ

Answercos 5θ − cos 9θ = 2 sin 7θ sin 2θ shown

Part (c) · 7 marks

Solve the equation for 0 < θ ≤ π/3, answers in terms of π.

  1. 12 sin 7θ sin 2θ = √3 sin 7θ
  2. 2sin 7θ(2 sin 2θ − √3) = 0
  3. 3sin 7θ = 0: 7θ = π, 2π → θ = π/7, 2π/7
  4. 4sin 2θ = √3/2: 2θ = π/3, 2π/3 → θ = π/6, π/3

Answerθ = π/7, 2π/7, π/6, π/3

Part (d) · 6 marks

Simplify the integrand and integrate from 0 to π/7.

  1. 1tan 2x cos 2x = sin 2x, so integrand = 8 sin 7x sin 2x = 4(2 sin 7x sin 2x)
  2. 2= 4(cos 5x − cos 9x)
  3. 3∫ = 4[sin 5x/5 − sin 9x/9] from 0 to π/7
  4. 4= 4(sin(5π/7)/5 − sin(9π/7)/9) − 0
  5. 5= 0.9729… = 0.973

Answer0.973

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