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WMA12/01 Jan 2024, Question 9: Harder Trig Equations

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 14 marks

The question, in short

Solve sin x tan x = 5 for 0 ≤ x < 360° to one decimal place, then use the curve y = A sin(2θ − 3π/8) + 2 with P's y-coordinate 7 to find A, Q and R.

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

Part (i) · 6 marks

Solve sin x tan x = 5 for 0 ≤ x < 360°, giving answers to one decimal place with full reasoning.

  1. 1sin x × (sin x/cos x) = 5, so sin²x = 5cos x
  2. 21 − cos²x = 5cos x, so cos²x + 5cos x − 1 = 0
  3. 3cos x = (−5 ± √29)/2
  4. 4(−5 − √29)/2 < −1 is impossible, so cos x = (−5 + √29)/2 ≈ 0.1926
  5. 5x = 78.9° or x = 360° − 78.9° = 281.1°

Answerx = 78.9°, 281.1°

Part (iia) · 1 mark

Given the y-coordinate of P is 7, state the value of the constant A.

  1. 1The maximum value is A + 2
  2. 2A + 2 = 7, so A = 5

AnswerA = 5

Part (iib) · 3 marks

Find the exact coordinates of the minimum point Q on the curve.

  1. 1Q is a minimum: sin(2θ − 3π/8) = −1, y = −5 + 2 = −3
  2. 2First minimum after P: 2θ − 3π/8 = 3π/2
  3. 32θ = 15π/8, so θ = 15π/16
  4. 4Q = (15π/16, −3)

AnswerQ(15π/16, −3)

Part (iic) · 4 marks

Find the value of θ at R, where the curve crosses the θ-axis, to 3 significant figures.

  1. 10 = 5 sin(2θ − 3π/8) + 2, so sin(2θ − 3π/8) = −2/5
  2. 2arcsin(−0.4) = −0.4115; other solutions are π + 0.4115, 2π − 0.4115, 3π + 0.4115, …
  3. 3R is the crossing going down after the second maximum: 2θ − 3π/8 = 3π + 0.4115 = 9.8363
  4. 42θ = 9.8363 + 1.1781 = 11.0144
  5. 5θ = 5.51 (3 s.f.)

Answerθ = 5.51

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