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4PM1/02R Jan 2022, Question 7: Trig Identities and Proofs

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 8 marks

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The question, in short

Using the given trigonometric equation, prove the stated value of cos θ, then apply that result to solve a related equation in 3x over the interval 0° ≤ x < 180°.

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

Part (a) · 4 marks

Rearrange and simplify the given fraction equation to demonstrate that cos θ equals −1/2.

  1. 1sin²θ = 1 − cos²θ, so 3 + sin²θ = 4 − cos²θ
  2. 24 − cos²θ = 3cos θ(cos θ − 2) = 3cos²θ − 6cos θ
  3. 34cos²θ − 6cos θ − 4 = 0
  4. 42(2cos θ + 1)(cos θ − 2) = 0
  5. 5cos θ = 2 is impossible, so cos θ = −1/2

Answercos θ = −1/2 shown

Part (b) · 4 marks

Use the result from part a to find all values of x satisfying the second equation for 0° ≤ x < 180°.

  1. 1cos 3x = −1/2
  2. 20° ≤ 3x < 540°
  3. 33x = 120°, 240°, 480°
  4. 4x = 40°, 80°, 160°

Answerx = 40°, 80°, 160°

Stuck on a step? Read the free lesson: Trig Identities and Proofs

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