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4PM1/02 Jan 2023, Question 10: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 13 marks

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The question, in short

Using given formulae, prove two double-angle identities, apply them to rewrite sin 2θ − tan θ as tan θ cos 2θ, then solve sin 2x° − tan x° = 0 for 0 < x < 360.

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

Part (a) · 5 marks

Show that sin 2θ = 2 sin θ cos θ and that cos 2θ = 2 cos²θ − 1, using the formulae on page 2.

  1. 1(i) sin 2θ = sin(θ + θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ
  2. 2(ii) cos 2θ = cos(θ + θ) = cos²θ − sin²θ
  3. 3= cos²θ − (1 − cos²θ)
  4. 4= 2 cos²θ − 1

Answer(i) sin 2θ = 2 sin θ cos θ shown (ii) cos 2θ = 2 cos²θ − 1 shown

Part (b) · 4 marks

Given θ ≠ 90° + 180°n for integer n, use part (a) results to show sin 2θ − tan θ equals tan θ cos 2θ.

  1. 1sin 2θ − tan θ = 2 sin θ cos θ − sin θ/cos θ
  2. 2= sin θ(2 cos θ − 1/cos θ)
  3. 3= sin θ (2 cos²θ − 1)/cos θ
  4. 4= tan θ cos 2θ

Answersin 2θ − tan θ = tan θ cos 2θ shown

Part (c) · 4 marks

Solve sin 2x° − tan x° = 0 for 0 < x < 360.

  1. 1tan x cos 2x = 0
  2. 2tan x = 0: x = 180 (0 and 360 are outside the range)
  3. 3cos 2x = 0: 2x = 90, 270, 450, 630
  4. 4x = 45, 135, 225, 315

Answerx = 45, 135, 180, 225, 315

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