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4PM1/02R Jun 2023, Question 11: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 12 marks

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The question, in short

Use a given formula to prove two trigonometric identities in sin²A and sin⁴x + cos⁴x, then apply them to solve an equation in θ for 0° ≤ θ < 180° to one decimal place.

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

Part (a) · 3 marks

Use a formula from page 2 to show that sin²A equals (1/2)(1 − cos 2A).

  1. 1cos 2A = cos(A + A) = cos²A − sin²A
  2. 2= (1 − sin²A) − sin²A = 1 − 2sin²A
  3. 32sin²A = 1 − cos 2A, so sin²A = (1/2)(1 − cos 2A)

Answersin²A = (1/2)(1 − cos 2A) shown

Part (b) · 5 marks

Show that sin⁴x + cos⁴x simplifies to (3 + cos 4x)/4.

  1. 1cos²x = (1/2)(1 + cos 2x) and sin²x = (1/2)(1 − cos 2x)
  2. 2sin⁴x + cos⁴x = (1/4)[(1 − cos 2x)² + (1 + cos 2x)²]
  3. 3= (1/4)(2 + 2cos²2x) = (1/2)(1 + cos²2x)
  4. 4cos²2x = (1 + cos 4x)/2
  5. 5= (1/2)(1 + (1 + cos 4x)/2) = (3 + cos 4x)/4

Answersin⁴x + cos⁴x = (3 + cos 4x)/4 shown

Part (c) · 4 marks

Hence solve 8sin⁴(θ/2) + 8cos⁴(θ/2) = 5sin(2θ) + 6 for 0° ≤ θ < 180°, giving θ in degrees to one decimal place.

  1. 1With x = θ/2: 8(sin⁴(θ/2) + cos⁴(θ/2)) = 8 × (3 + cos 2θ)/4 = 6 + 2cos 2θ
  2. 26 + 2cos 2θ = 5 sin 2θ + 6, so 5 sin 2θ = 2 cos 2θ
  3. 3tan 2θ = 2/5
  4. 42θ = 21.80°, 201.80°
  5. 5θ = 10.9°, 100.9°

Answerθ = 10.9°, 100.9°

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