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4PM1/01 Jun 2023, Question 9: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 13 marks

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The question, in short

Use given formulae to derive two half-angle identities, verify a product-to-double-angle identity, then solve dy/dx = 0 for 0° ≤ x ≤ 180° to the nearest whole number.

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

Part (a) · 4 marks

Starting from the formulae provided, derive the identities for cos²A and sin²A in terms of cos 2A.

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

Answercos²A = (cos 2A + 1)/2 and sin²A = (1 − cos 2A)/2 shown

Part (b) · 5 marks

Expand the product of the two brackets and use double-angle identities to show it equals (1/2)(cos 2x − 7 sin 2x + 5).

  1. 1Expand: 2sin²x − 6sin x cos x − sin x cos x + 3cos²x = 2sin²x − 7sin x cos x + 3cos²x
  2. 22sin²x = 2 × (1 − cos 2x)/2 = 1 − cos 2x
  3. 33cos²x = 3 × (cos 2x + 1)/2
  4. 47sin x cos x = (7/2) sin 2x
  5. 5Total: 1 − cos 2x + (3/2)cos 2x + 3/2 − (7/2)sin 2x = (1/2)(cos 2x − 7 sin 2x + 5)

Answer(2 sin x − cos x)(sin x − 3 cos x) = (1/2)(cos 2x − 7 sin 2x + 5) shown

Part (c) · 4 marks

Differentiate y, set dy/dx to zero, and find x in 0° ≤ x ≤ 180° to the nearest whole number.

  1. 1y = (1/2)(cos 2x − 7 sin 2x + 5)
  2. 2dy/dx = −sin 2x − 7cos 2x = 0
  3. 3tan 2x = −7
  4. 42x = 98.13°, 278.13°
  5. 5x = 49°, 139°

Answerx = 49°, 139°

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