WMA12/01 Jan 2025, Question 10: Trig Identities and Proofs
Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks
The question, in short
Show a trigonometric identity by manipulating tan and cos into sin terms, then use it to solve an equation for x between 0 and 2π.
Summarised in our own words. Use the official paper for the exact wording and any diagrams.
Part (a) · 4 marks
Show the identity linking cos θ times a bracket of 3 tan θ plus 2 over tan θ to sin θ plus 2 over sin θ.
- 1Use tan θ = sin θ/cos θ: cos θ(3 sin θ/cos θ + 2 cos θ/sin θ)
- 2= 3 sin θ + 2cos²θ/sin θ
- 3Use cos²θ = 1 − sin²θ: = 3 sin θ + 2(1 − sin²θ)/sin θ
- 4= 3 sin θ + 2/sin θ − 2 sin θ = sin θ + 2/sin θ
AnswerShown: cos θ(3 sin θ/cos θ + 2 cos θ/sin θ) = 3 sin θ + 2(1 − sin²θ)/sin θ = sin θ + 2/sin θ
Part (b) · 4 marks
Use the identity to solve the equation equal to 4 sin x − 5 for 0 < x < 2π, answers to 3 significant figures.
- 1Using (a): sin x + 2/sin x = 4 sin x − 5
- 2Multiply by sin x: sin²x + 2 = 4 sin²x − 5 sin x
- 33 sin²x − 5 sin x − 2 = 0, so (3 sin x + 1)(sin x − 2) = 0
- 4sin x = 2 has no solutions, so sin x = −1/3
- 5x = π + 0.3398… = 3.48 or x = 2π − 0.3398… = 5.94
Answerx = 3.48, 5.94
Stuck on a step? Read the free lesson: Trig Identities and Proofs
Practise more trig identities and proofs questions
Real XMA01 past-paper questions, marked on your working like the exam, with a worked solution for every part.
Practise free on LemmaraMore past-paper questions on trig identities and proofs
Also searched as: WMA12 January 2025 · WMA12/01 · wma12 jan25 · IAS Maths Pure Mathematics 2 January 2025 · question 10.
Lemmara is an independent study tool, not affiliated with or endorsed by Pearson Edexcel. The worked solutions are Lemmara’s own.