0580/43 Oct/Nov 2023, Question 7: Trigonometric Graphs and Equations

Part (a) · 3 marks

Work out the missing y values for y = 3cos 2x° at x = 45, 60 and 80, giving each to 1 decimal place.

1. 1x = 45: 3cos 90° = 0
2. 2x = 60: 3cos 120° = −1.5
3. 3x = 80: 3cos 160° = −2.819… ≈ −2.8

Answer0, −1.5, −2.8

Part (b) · 4 marks

Plot the cosine curve y = 3cos 2x° on the grid for x from 0 to 90.

1. 1Plot (0, 3), (10, 2.8), (20, 2.3), (30, 1.5), (40, 0.5), (45, 0), (50, −0.5), (60, −1.5), (70, −2.3), (80, −2.8), (90, −3)
2. 2Join with a smooth curve

AnswerCorrect graph

Part (c) · 1 mark

Use the drawn curve to find x where 3cos 2x° equals −2 between 0 and 90.

1. 1Draw y = −2 and read the x-value where it meets the curve
2. 2(Check: cos 2x = −2/3 gives 2x = 131.8…, x = 65.9…)
3. 3x ≈ 66

Answer65 to 67

Part (d) · 3 marks

Sketch y = 3cos 2x°, add a suitable straight line, and solve 120cos 2x° = 80 − x for 0 ≤ x ≤ 90.

1. 1Divide by 40: 3cos 2x = 2 − x/40
2. 2Draw the line y = 2 − x/40, through (0, 2) and (80, 0)
3. 3Read the x-value where the line meets the curve
4. 4x ≈ 34

Answer32 to 36

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