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4PM1/01R Jun 2025, Question 10: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 15 marks

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The question, in short

Use a given formula to derive two double-angle identities, find where y = sin x meets y = cos 2x, then find the exact volume when region R is rotated 360° about the x-axis.

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

Part (a) · 5 marks

Starting from a formula on page 2, derive the identities for sin²A and cos²A in terms of cos 2A.

  1. 1(i) cos(A + A) = cos A cos A − sin A sin A, so cos 2A = cos²A − sin²A
  2. 2cos 2A = 1 − sin²A − sin²A = 1 − 2sin²A
  3. 32sin²A = 1 − cos 2A
  4. 4sin²A = (1 − cos 2A)/2
  5. 5(ii) cos 2A = 1 − 2sin²A = 1 − 2(1 − cos²A)
  6. 6cos 2A = 2cos²A − 1
  7. 7cos²A = (cos 2A + 1)/2

Answer(i) sin²A = (1 − cos 2A)/2 shown; (ii) cos²A = (cos 2A + 1)/2 shown

Part (b) · 5 marks

Solve algebraically for the exact coordinates of the intersection point A of the two curves.

  1. 1cos 2x = sin x
  2. 21 − 2sin²x = sin x
  3. 32sin²x + sin x − 1 = 0
  4. 4(2sin x − 1)(sin x + 1) = 0
  5. 5sin x = 1/2, so x = π/6 and y = 1/2

AnswerA(π/6, 1/2)

Part (c) · 5 marks

Integrate algebraically to get the exact volume of revolution of R about the x-axis, in the form (a√a/b)π.

  1. 1V = π ∫ from 0 to π/6 of (cos²2x − sin²x) dx
  2. 2= π ∫ from 0 to π/6 of [(cos 4x + 1)/2 − (1 − cos 2x)/2] dx
  3. 3= π ∫ from 0 to π/6 of (cos 4x/2 + cos 2x/2) dx
  4. 4= π [sin 4x/8 + sin 2x/4] from 0 to π/6
  5. 5= π(√3/16 + √3/8) = (3√3/16)π

Answer(3√3/16)π

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