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WMA11/01 Jan 2025, Question 8: The Cosine Rule

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 13 marks

The question, in short

For a badge formed from triangle OAB plus sector OBC centred at O, with OB = 3.4 cm, AB = 1.9 cm and angle AOB = π/6, work out angles, areas and the badge perimeter.

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

Part (a) · 3 marks

Using the given sides and angle, determine angle OAB in radians, rounded to 4 significant figures, showing your method.

  1. 1Sine rule: sin OAB/3.4 = sin(π/6)/1.9
  2. 2sin OAB = 3.4 × 0.5/1.9 = 0.8947…
  3. 3arcsin(0.8947…) = 1.1071 (acute)
  4. 4OAB is obtuse, so OAB = π − 1.1071 = 2.034 (4 s.f.)

AnswerAngle OAB = 2.034 radians

Part (b) · 2 marks

Calculate the area of triangle OAB in cm², correct to 3 significant figures.

  1. 1Angle OBA = π − π/6 − 2.0338 = 0.5842
  2. 2Area = ½ × 3.4 × 1.9 × sin(0.5842)
  3. 3= 3.23 × 0.5516 = 1.78 cm² (3 s.f.)

Answer1.78 cm²

Part (c) · 3 marks

Given sector OBC area to triangle OAB area is 3 : 2, demonstrate that angle BOC equals 0.462 radians to 3 significant figures.

  1. 1Sector area = (3/2) × 1.7816 = 2.6723 cm²
  2. 2½ × 3.4² × θ = 2.6723
  3. 35.78θ = 2.6723
  4. 4θ = 0.4623… = 0.462 radians (3 s.f.)

AnswerShown: sector area = 1.5 × 1.7816 = 2.672; ½ × 3.4² × θ = 2.672 ⇒ θ = 0.462

Part (d) · 5 marks

Using your previous results, work out the total perimeter of the badge in cm, rounded to the nearest integer.

  1. 1OA² = 3.4² + 1.9² − 2 × 3.4 × 1.9 × cos(0.5842)
  2. 2OA = 2.096 cm
  3. 3Arc BC = rθ = 3.4 × 0.4623 = 1.572 cm
  4. 4Perimeter = OA + AB + arc BC + OC = 2.096 + 1.9 + 1.572 + 3.4
  5. 5= 8.97 ≈ 9 cm

Answer9 cm

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