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4PM1/02 Jun 2024, Question 11: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 16 marks

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The question, in short

Use given formulae to prove two double-angle identities, a triple-angle cosine identity, then solve a trigonometric equation and evaluate a definite integral.

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

Part (ai) · 3 marks

Starting from the given formulae, prove that cos 2A equals 2cos²A − 1.

  1. 1cos 2A = cos(A + A) = cos²A − sin²A
  2. 2= cos²A − (1 − cos²A)
  3. 3= 2cos²A − 1

Answercos 2A = 2cos²A − 1 shown

Part (aii) · 1 mark

Using the given formulae, prove that sin 2A equals 2 sin A cos A.

  1. 1sin 2A = sin(A + A) = sin A cos A + cos A sin A
  2. 2= 2 sin A cos A

Answersin 2A = 2 sin A cos A shown

Part (b) · 4 marks

Show that cos³A can be written as (cos 3A + 3cos A) ÷ 4.

  1. 1cos 3A = cos(2A + A) = cos 2A cos A − sin 2A sin A
  2. 2= (2cos²A − 1)cos A − 2 sin²A cos A
  3. 3= 2cos³A − cos A − 2(1 − cos²A)cos A
  4. 4= 4cos³A − 3cos A
  5. 5So cos³A = (cos 3A + 3cos A)/4

Answercos³A = (cos 3A + 3cos A)/4 shown

Part (c) · 4 marks

Find exact solutions in terms of π for 8cos³(θ/2) − 6cos(θ/2) − 1 = 0 with 0 ≤ θ ≤ 2π.

  1. 18cos³(θ/2) − 6cos(θ/2) = 2cos(3θ/2)
  2. 2So 2cos(3θ/2) − 1 = 0, cos(3θ/2) = 1/2
  3. 33θ/2 = π/3, 5π/3, 7π/3 (11π/3 gives θ > 2π)
  4. 4θ = 2π/9, 10π/9, 14π/9

Answerθ = 2π/9, 10π/9, 14π/9

Part (d) · 4 marks

Evaluate exactly, by algebraic integration, the integral from 0 to π/6 of (4cos³θ − sin 2θ) dθ.

  1. 14cos³θ = cos 3θ + 3cos θ
  2. 2∫ (cos 3θ + 3cos θ − sin 2θ) dθ = sin 3θ/3 + 3 sin θ + cos 2θ/2
  3. 3At π/6: 1/3 + 3/2 + 1/4; at 0: 1/2
  4. 419/12

Answer19/12

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