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4PM1/01 Jan 2022, Question 6: The Sine Rule

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

Triangle ABC has AC = x cm, AB = (x + 3) cm, angle ABC = 30° and angle ACB = θ° with 0 < θ < 90. Derive two expressions for sin θ° and cos θ°, then use a given angle ratio to find x exactly.

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

Part (a) · 5 marks

Using the triangle's side lengths and the 30° angle, derive the stated expressions for sin θ° and cos θ° in terms of x.

  1. 1Sine rule: sin 30°/x = sin θ°/(x + 3)
  2. 2sin θ° = (x + 3)/(2x)
  3. 3cos²θ° = 1 − (x + 3)²/(4x²) = (4x² − (x² + 6x + 9))/(4x²)
  4. 4= (3x² − 6x − 9)/(4x²)
  5. 5θ is acute, so cos θ° = √(3x² − 6x − 9)/(2x)

Answersin θ° = (x + 3)/(2x) and cos θ° = √(3x² − 6x − 9)/(2x) shown

Part (b) · 5 marks

Given angle BAC to angle ABC is 7 : 2, determine x exactly as a + a√b with a and b prime.

  1. 1∠BAC/30° = 7/2, so ∠BAC = 105°
  2. 2θ = 180 − 30 − 105 = 45
  3. 3cos 45° = √2/2 = √(3x² − 6x − 9)/(2x)
  4. 42x² = 3x² − 6x − 9, so x² − 6x − 9 = 0
  5. 5(x − 3)² = 18, x = 3 + 3√2 (positive root)

Answerx = 3 + 3√2

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