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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Trigonometric graphs and exact values

Graphs of sin, cos and tan, transformations, exact values at 30, 45 and 60 degrees.

12 original questions, 47 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

Without using a calculator, find the exact value of

sin 60° cos 30° + tan 45°

2 marks
Show answer
7/4
2

Without using a calculator, find the exact value of

  1. cos 150°
  2. tan 225°
  3. sin 300°
3 marks
Show answer
a) -√3/2 b) 1 c) -√3/2
3

The point P(90°, 1) lies on the curve with equation y = sin x

Write down the coordinates of the point to which P is mapped on the curve with equation

  1. y = 3 sin x
  2. y = sin(x − 60°)
  3. y = sin 2x
  4. y = sin x − 2
4 marks
Show answer
a) (90°, 3) b) (150°, 1) c) (45°, 1) d) (90°, -1)
4

The diagram shows part of the curve with equation y = a sin bx, where a and b are positive constants and x is in degrees.

The curve has a maximum point at (45°, 4) and crosses the positive x-axis for the first time at (90°, 0).

Find the value of a and the value of b.

xy(45°, 4)(90°, 0)
3 marks
Show answer
a = 4, b = 2
5

On the axes provided, sketch the graph of y = sin(x + 30°) for 0 ≤ x ≤ 360°

Give the coordinates of each point where the graph meets the coordinate axes.

090180270360−1.5−1−0.50.511.5xy
4 marks
Show answer
Sine curve translated 30° left: meets the y-axis at (0, 1/2) and the x-axis at (150°, 0) and (330°, 0); maximum (60°, 1), minimum (240°, -1).
6
  1. Describe fully the single transformation that maps the graph of y = tan x onto the graph of y = tan(x − 45°)
  2. Sketch the graph of y = tan(x − 45°) for 0 ≤ x ≤ 360°, showing the equations of any asymptotes and the coordinates of the points where the graph meets the coordinate axes.
04590135180225270315360−4−3−2−11234xy
4 marks
Show answer
a) Translation 45° in the positive x-direction b) Asymptotes x = 135° and x = 315°; (0, -1), (45°, 0), (225°, 0)
7

The curve with equation y = a + b cos x, where a and b are constants and b > 0, has a maximum value of 7 and a minimum value of −1

  1. Find the value of a and the value of b.
  2. Write down the coordinates of the minimum point of the curve for 0 ≤ x ≤ 360°
4 marks
Show answer
a) a = 3, b = 4 b) (180°, -1)
8

Without using a calculator, show that

1 + tan 60°1 − tan 60° = −2 − √3

4 marks
Show answer
tan 60° = √3, and (1 + √3)/(1 - √3) = (1 + √3)2/(1 - 3) = (4 + 2√3)/(-2) = -2 - √3.
9
  1. On the same axes, sketch the graphs of y = sin 2x and y = cos x for 0 ≤ x ≤ 360°
  2. Hence state the number of solutions of the equation sin 2x = cos x in the interval 0 ≤ x ≤ 360°, giving a reason for your answer.
090180270360−1.5−1−0.50.511.5xy
4 marks
Show answer
a) sin 2x: period 180°, max 1 at 45° and 225°, min -1 at 135° and 315°; cos x: from 1 at 0°, -1 at 180°, back to 1 at 360° b) 4 solutions: the curves intersect 4 times
10

The diagram shows part of the curve with equation y = sin(x + α), where 0 < α < 90°

The curve crosses the x-axis at (120°, 0) and (300°, 0) and crosses the y-axis at the point P.

  1. Find the value of α.
  2. Find the exact y-coordinate of P.
xyP(120°, 0)(300°, 0)
4 marks
Show answer
a) α = 60° b) √3/2
11

Given that sin 20° = p, find, in terms of p

  1. sin 160°
  2. sin 340°
  3. cos 70°
  4. tan 20°
5 marks
Show answer
a) p b) -p c) p d) p/√(1 - p2)
12

The depth of water, D metres, at the entrance to a harbour in Cornwall is modelled by

D = 6 + 2.5 sin(30t)°,   0 ≤ t < 24

where t is the time in hours after midnight.

  1. Find the maximum depth of the water and the first time after midnight at which it occurs.
  2. State the time, in hours, between one high tide and the next.
  3. A fishing boat can only pass the entrance when the depth is at least 7.25 m. Find the first period of time after midnight during which the boat can pass.
6 marks
Show answer
a) 8.5 m at 03:00 b) 12 hours c) 01:00 to 05:00

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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