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WMA12/01 Jun 2019, Question 9: Trig Identities and Proofs

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Show a trigonometric equation can be rearranged into a given quadratic in cosθ, then use that result to solve a related equation in x over 0 ≤ x < π/2, giving answers to 2 decimal places.

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

Part (a) · 4 marks

Show that cosθ − 1 = 4sinθ tanθ rearranges to 5cos²θ − cosθ − 4 = 0.

  1. 1Use tanθ = sinθ/cosθ: cosθ − 1 = 4sin²θ/cosθ
  2. 2Multiply by cosθ: cos²θ − cosθ = 4sin²θ
  3. 3Use sin²θ = 1 − cos²θ: cos²θ − cosθ = 4 − 4cos²θ
  4. 45cos²θ − cosθ − 4 = 0

AnswerShown: cosθ − 1 = 4sin²θ/cosθ ⇒ cos²θ − cosθ = 4(1 − cos²θ) ⇒ 5cos²θ − cosθ − 4 = 0

Part (b) · 4 marks

Hence solve cos2x − 1 = 4sin2x tan2x for 0 ≤ x < π/2, giving answers to 2 decimal places where appropriate.

  1. 1With θ = 2x: 5cos²2x − cos2x − 4 = 0
  2. 2(5cos2x + 4)(cos2x − 1) = 0
  3. 3cos2x = 1 or cos2x = −4/5
  4. 4cos2x = 1 ⇒ 2x = 0 ⇒ x = 0
  5. 5cos2x = −4/5 ⇒ 2x = 2.498 (the other value 3.785 gives x > π/2)
  6. 6x = 1.25

Answerx = 0, 1.25

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Also searched as: WMA12 June 2019 · WMA12/01 · wma12 jun19 · IAS Maths Pure Mathematics 2 June 2019 · question 9.
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