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4PM1/02 Jun 2022, Question 7: The Addition Formulae

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 11 marks

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The question, in short

Use page 2 formulae to prove a double-angle tangent identity, find integers a and b in a surd form for tan α, rewrite a trigonometric equation as tan x° = 1, then solve a related equation for y in −90 < y ≤ 90.

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

Part (ia) · 2 marks

Starting from a page 2 formula, prove that tan 2θ equals 2tan θ divided by 1 − tan²θ.

  1. 1tan 2θ = tan(θ + θ) = (tan θ + tan θ)/(1 − tan θ tan θ)
  2. 2= 2tan θ/(1 − tan²θ)

Answertan 2θ = 2tan θ/(1 − tan²θ) shown

Part (ib) · 3 marks

Given tan 2α = 1, derive tan α in the form a ± √b and state the integer values of a and b.

  1. 11 = 2tan α/(1 − tan²α)
  2. 21 − tan²α = 2tan α ⇒ tan²α + 2tan α − 1 = 0
  3. 3tan α = (−2 ± √8)/2 = −1 ± √2

Answertan α = −1 ± √2 (a = −1, b = 2)

Part (iia) · 4 marks

Using page 2 formulae, show that cos(x − 30)° = sin(x + 30)° simplifies to tan x° = 1.

  1. 1cos x cos 30° + sin x sin 30° = sin x cos 30° + cos x sin 30°
  2. 2cos x(cos 30° − sin 30°) = sin x(cos 30° − sin 30°)
  3. 3(√3/2 − 1/2)cos x = (√3/2 − 1/2)sin x
  4. 4sin x/cos x = 1 ⇒ tan x° = 1

Answertan x° = 1 shown

Part (iib) · 2 marks

Hence or otherwise find all y with −90 < y ≤ 90 satisfying cos(2y − 30)° = sin(2y + 30)°.

  1. 1Replace x by 2y: tan 2y = 1
  2. 2−180 < 2y ≤ 180: 2y = 45 or 2y = −135
  3. 3y = 22.5 or y = −67.5

Answery = −67.5, 22.5

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