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WMA11/01 Jun 2022, Question 9: Trig Graphs: Shifts and Counting Solutions

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 8 marks

The question, in short

Given sin α = p with 0 < α < 90°, express related sine values in terms of p, sketch y = sin 2x on the given sine curve figure, then use it to find x-coordinates where sin 2x = p for 0 < x < 180°.

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

Part (ai) · 1 mark

State 2 sin(180° − α) in terms of p.

  1. 1sin(180° − α) = sin α = p
  2. 22 sin(180° − α) = 2p

Answer2p

Part (aii) · 1 mark

State sin(α − 180°) in terms of p.

  1. 1The sine graph gives sin(α − 180°) = −sin α
  2. 2So sin(α − 180°) = −p

Answer−p

Part (aiii) · 1 mark

State 3 + sin(180° + α) in terms of p.

  1. 1sin(180° + α) = −sin α = −p
  2. 23 + sin(180° + α) = 3 − p

Answer3 − p

Part (b) · 2 marks

On a copy of the given sine graph, sketch y = sin 2x.

  1. 1y = sin 2x is a horizontal stretch of y = sin x by scale factor ½
  2. 2Same amplitude 1, period 180°, crossing the x-axis every 90°

AnswerSketch: a sine wave with the same height (between −1 and 1) through O, but twice as many waves — period 180°, crossing the x-axis at 0, ±90°, ±180°, ±270°, ±360°.

Part (c) · 3 marks

Using your sketch, find in terms of α the x-coordinates in 0 < x < 180° where sin 2x = p.

  1. 1sin 2x = p = sin α, so 2x = α or 2x = 180° − α (for 0 < 2x < 360°)
  2. 2x = α/2
  3. 3x = (180° − α)/2 = 90° − α/2

Answerx = α/2 and x = 90° − α/2

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