A LevelMathsCambridge (CIE)Pure 1Exam QuestionsTrigonometryTrigonometric Functions

Trigonometric Functions (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

Exam code: 9709

3 hours42 questions
Easy
Medium
Hard
Very Hard
1
6 marks

On separate diagrams, sketch the graphs of:

(i) y = \text{sin} \; x for -180^{\circ} \leq x \leq 180^{\circ}

(ii) y = \text{cos} \; x for 0^{\circ} \leq x \leq 360^{\circ}

(iii) y = \text{tan} \; x for -180^{\circ} \leq x \leq 180^{\circ}

How did you do?

2
3 marks

Sketch the graph of y = \text{sin} \; 2x for 0^{\circ} \leq x \leq 180^{\circ}.

How did you do?

3
2 marks

(i) Write down the maximum value of y, where y = 3 \, \text{cos} \; x.

(ii) Write down the minimum value of y, where y = 9 \, \text{sin} \; x.

How did you do?

4
3 marks

The point P has coordinates \left(90^{\circ},\ 1\right) and lies on the graph of y = \text{f}(x), where \text{f}(x) = \text{sin} \; x and 0^{\circ} \leq x \leq 180^{\circ}.

Write down the coordinates of the image of point P under each of the following transformations:

(i) y = \text{f}(x) + 2

(ii) y = \text{f}(3x)

(iii) y = \text{f}(x + 30^{\circ})

How did you do?

5
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2 marks

Solve \text{cos} \; x = \dfrac{1}{2} for 0^{\circ} \leq x \leq 360^{\circ}.

How did you do?

6
2 marks

The diagram shows the graph of y = \text{tan} \; x for 0^{\circ} \leq x \leq 720^{\circ}.

By adding a suitable straight line to the graph, show that the equation \text{tan} \; x = 2 has four solutions in the interval 0^{\circ} \leq x \leq 720^{\circ}.

Graph of y = tan x for 0° ≤ x ≤ 720°

How did you do?

7
2 marks

Sketch the graph of y = -\text{sin} \; \theta for 0^{\circ} \leq \theta \leq 360^{\circ}.

How did you do?

8
3 marks

Given that \text{f}(\theta) = \text{cos} \; \theta, write each of the following as an expression involving the cosine function.

(i) 2 \, \text{f}(\theta) + 3

(ii) 3 \, \text{f}(2\theta)

How did you do?

9
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2 marks

Solve \text{sin} \; 3x = 0 for 0^{\circ} \leq x \leq 360^{\circ}.

How did you do?

1
3 marks

(i) Sketch the graph of y = \text{cos} \; \theta in the interval 0^{\circ} \leq \theta \leq 360^{\circ}. The sketch must include the coordinates of all points where the graph meets the coordinate axes.

(ii) Write down all the values of \theta for which \text{cos} \; \theta = 0 for 0^{\circ} \leq \theta \leq 360^{\circ}.

How did you do?

2
3 marks

(i) Sketch the graph of y = \text{sin} \; \theta in the interval 0^{\circ} \leq \theta \leq 360^{\circ}. The sketch must include the coordinates of all points where the graph meets the coordinate axes.

(ii) Given that \text{sin} \; 30^{\circ} = 0.5, use your graph to find another value of \theta in the given range for which \text{sin} \; \theta = 0.5.

How did you do?

3
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3 marks

By sketching an appropriate graph, find all the solutions of \text{tan} \; \theta = -1 in the interval 0^{\circ} \leq \theta \leq 360^{\circ}.

How did you do?

4
4 marks

(i) Sketch the graph of y = \text{cos}(\theta + 30^{\circ}) in the interval -180^{\circ} \leq \theta \leq 360^{\circ}.

(ii) Write down all the values where \text{cos}(\theta + 30^{\circ}) = 0 in the given interval.

How did you do?

5
5 marks

On the same set of axes, sketch the graphs of the following functions:

(i) y = 2 \, \text{sin} \; \theta in the interval 0^{\circ} \leq \theta \leq 360^{\circ}

(ii) y = -2 \, \text{sin} \; \theta in the interval 0^{\circ} \leq \theta \leq 360^{\circ}

The sketch must include the coordinates of all points where the graphs meet the coordinate axes. Also state the periodicity of each function.

How did you do?

6
5 marks

(i) On the same set of axes, sketch the graphs of y = \text{cos} \; \theta and y = \text{cos} \; 3\theta in the interval -180^{\circ} \leq \theta \leq 180^{\circ}, giving the coordinates of all points of intersection with the coordinate axes.

(ii) State the coordinates of any points where the graphs intersect.

How did you do?

7a
2 marks

The graph shows the curve with equation y = \text{tan}(x + 50^{\circ}) in the interval -180^{\circ} \leq x \leq 360^{\circ}.

Graph of y = tan(x + 50°) for −180° ≤ x ≤ 360°

A student states that the curve could also have equation y = \text{tan}(x - 130^{\circ}).

Is the student correct? You must give a reason for your answer.

How did you do?

7b
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2 marks

Give the coordinates of all points of intersection with the coordinate axes within the interval.

How did you do?

7c
1 mark

Give another example of an equation that would also produce the same curve.

How did you do?

8a
1 mark

The graph shows the curve with equation y = \text{sin} \; 2x in the interval -60^{\circ} \leq x \leq 270^{\circ}.

Point A has coordinates \left(-45^{\circ},\ -1\right) and is the minimum point closest to the origin. Point B is the maximum point closest to the origin.

Graph of y = sin 2x for −60° ≤ x ≤ 270°, with the line y = −½; A is the nearest minimum (−45°, −1), P is the point (−15°, −½), and B, Q, R are further points marked with letters only

State the coordinates of B.

How did you do?

8b
2 marks

A straight line with equation y = -\dfrac{1}{2} meets the graph of y = \text{sin} \; 2x at the three points P, Q and R, as shown in the diagram.

Graph of y = sin 2x for −60° ≤ x ≤ 270°, with the line y = −½; A is the nearest minimum (−45°, −1), P is the point (−15°, −½), and B, Q, R are further points marked with letters only

Given that point P has coordinates \left(-15^{\circ},\ -\dfrac{1}{2}\right), use graph symmetries to determine the coordinates of Q and R.

How did you do?

9
3 marks

(i) Describe geometrically the transformation that maps the graph of y = \text{cos} \; x onto the graph of y = 4 \, \text{cos} \; x.

(ii) On the graph of y = \text{cos} \; x, a point P has coordinates \left(60^{\circ},\ 0.5\right). State the new coordinates of point P after the transformation to y = 4 \, \text{cos} \; x.

How did you do?

10
3 marks

(i) Describe geometrically the transformation that maps the graph of y = \text{sin} x onto the graph of y = \text{sin} 3 x.

(ii) On the graph of y = \text{sin} x, a point Q has coordinates \left(60 \circ , \frac{\sqrt{3}}{2}\right). State the new coordinates of point Q after the transformation to y = \text{sin} 3 x.

How did you do?

11
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6 marks

A section of a new rollercoaster has a series of rises and falls. The vertical displacement of the rollercoaster carriage, y, measured in metres relative to a fixed reference height, can be modelled using the function y = 30 \, \text{cos}(24t)^{\circ}, where t is the time in seconds.

(i) Sketch the function for the interval 0 \leq t \leq 30.

(ii) How many times will the rollercoaster carriage fall during the 30 seconds?

(iii) How long does the model suggest it will take for the rollercoaster carriage to reach the bottom of the first fall?

How did you do?

12
8 marks

(i) On the same set of axes, sketch the graphs of y = \text{sin} \; 2\theta and y = \text{cos}(\theta + 90^{\circ}) in the interval -180^{\circ} \leq \theta \leq 180^{\circ}. Show clearly the coordinates of all points of intersection with the coordinate axes.

(ii) Deduce the number of solutions to the equation \text{sin} \; 2\theta = \text{cos}(\theta + 90^{\circ}) in the interval -180^{\circ} \leq \theta \leq 180^{\circ}.

How did you do?

13
2 marks

(i) Sketch the graph of y = \text{cos} \; \theta in the interval -90^{\circ} \leq \theta \leq 360^{\circ}. The sketch must include the coordinates of all points where the graph meets the coordinate axes.

(ii) Given that \text{cos} \; 60^{\circ} = 0.5, use your graph to find all other values of \theta in the given interval for which \text{cos} \; \theta = 0.5.

How did you do?

14
3 marks

(i) Describe geometrically the transformation that maps the graph of y = \text{tan} \; x onto the graph of y = \dfrac{1}{5} \text{tan} \; x.

(ii) On the graph of y = \text{tan} \; x, a point Q has coordinates \left(30^{\circ},\ \dfrac{\sqrt{3}}{3}\right). State the new coordinates of point Q after the transformation to y = \dfrac{1}{5} \text{tan} \; x. Leave your answer in surd form.

How did you do?

1
3 marks

Sketch the graph of y = \text{tan} \; \theta in the interval -270^{\circ} \leq \theta \leq 270^{\circ}. The sketch must include the coordinates of all points where the graph meets the coordinate axes.

Given that \text{tan} \; 30^{\circ} = \dfrac{1}{\sqrt{3}}, use your graph to find all other values of \theta in the given interval for which \text{tan} \; \theta = \dfrac{1}{\sqrt{3}}.

How did you do?

2
3 marks

(i) Sketch the graph of y = \text{sin} \; \theta in the interval -180^{\circ} \leq \theta \leq 180^{\circ}. The sketch must include the coordinates of all points where the graph meets the coordinate axes.

(ii) Given that \text{sin} \; 60^{\circ} = \dfrac{\sqrt{3}}{2}, use your graph to find all values of \theta in the given interval for which \text{sin} \; \theta = -\dfrac{\sqrt{3}}{2}.

How did you do?

3
4 marks

Sketch the graph of y = \text{tan}(\theta - 45^{\circ}) in the interval -360^{\circ} \leq \theta \leq 360^{\circ}.

Write down all the values of \theta for which \text{tan}(\theta - 45^{\circ}) = 1 in the given interval.

How did you do?

4
6 marks

On the same set of axes, sketch the graphs of y = -3 \, \text{cos} \; \theta and y = \text{cos} \; 3\theta in the interval 0^{\circ} \leq \theta \leq 360^{\circ}.

The sketches must include the coordinates of all points where the graphs meet the coordinate axes. In each case state the periodicity of the function.

How did you do?

5
8 marks

(i) On the same set of axes, sketch the graphs of y = \dfrac{1}{2} \text{sin} \; \theta and y = \text{sin}(\theta - 60^{\circ}) in the interval -180^{\circ} \leq \theta \leq 180^{\circ}. State the coordinates of all points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

(ii) Verify that \theta = 90^{\circ} is a solution to the equation \dfrac{1}{2} \text{sin} \; \theta = \text{sin}(\theta - 60^{\circ}). Hence, using your sketch from part (i) or otherwise, find all other solutions to the equation \dfrac{1}{2} \text{sin} \; \theta = \text{sin}(\theta - 60^{\circ}) in the interval -180^{\circ} \leq \theta \leq 180^{\circ}.

How did you do?

6a
2 marks

The graph shows a curve with equation y = \text{cos}(x + k^{\circ}), -360^{\circ} \leq x \leq 360^{\circ}, where k is a constant.

Graph of y = cos(x + k°) for −360° ≤ x ≤ 360°

A student states that there is only one possible value for k. Explain why the student is incorrect, stating at least two possible values for k.

How did you do?

6b
2 marks

Give the coordinates of all points of intersection with the x-axis in the given interval.

How did you do?

7a
1 mark

The graph shows a curve with equation y = \text{sin} \; 3x in the interval -60^{\circ} \leq x \leq 150^{\circ}.

Graph of y = sin 3x for −60° ≤ x ≤ 150°, showing stationary points A and B nearest the origin

Points A and B are the stationary points closest to the origin. State the coordinates of A and B.

How did you do?

7b
2 marks

A straight line with equation y = \dfrac{\sqrt{3}}{2} meets the graph y = \text{sin} \; 3x at three points, R, S and T. Determine the coordinates of R, S and T.

How did you do?

8
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5 marks

Changes in the depth of water in a small tidal estuary relative to a fixed reference depth can be modelled using the function y = \text{sin}(22.5t)^{\circ}, where y is measured in metres and t is the time in hours.

(i) Sketch the function for the interval 0 \leq t \leq 8.

(ii) If t = 0 represents 2pm, during what times, to the nearest half hour, will the estuary be at or above the halfway point between y = 0 and its maximum depth?

How did you do?

9
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5 marks

A series of dips and mounds caused by underground mining has a cross-section which can be modelled using the function y = 4 \, \text{cos}(18x)^{\circ}, where x and y are respectively the horizontal and vertical displacements, in metres, from a fixed origin point.

(i) Sketch the function for the interval 0 \leq x \leq 40 and state the periodicity of the model.

(ii) How many dips are in this model in the given interval?

How did you do?

10
8 marks

(i) On the same set of axes, sketch the graphs of y = \text{tan} \; \dfrac{1}{4}\theta and y = \text{cos}(\theta + 120^{\circ}) in the interval 0^{\circ} \leq \theta \leq 270^{\circ}. Show clearly the coordinates of all points of intersection with the coordinate axes.

(ii) Deduce the number of solutions to the equation \text{cos}(\theta + 120^{\circ}) - \text{tan} \; \dfrac{1}{4}\theta = 0, in the interval 0^{\circ} \leq \theta \leq 270^{\circ}.

How did you do?

11
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4 marks

By sketching an appropriate graph, find all the solutions to \text{tan} \; \theta = \dfrac{-1}{\sqrt{3}} in the interval 0^{\circ} \leq \theta \leq 360^{\circ}.

How did you do?

1
6 marks

(i) On the same set of axes, sketch the graphs of y = \text{cos}(-2\theta) and y = \text{cos} \; \dfrac{1}{2}\theta in the interval -360^{\circ} \leq \theta \leq 360^{\circ}. Label the axes appropriately to show all points of intersection between the graphs and the coordinate axes.

(ii) State the periodicity of each function.

How did you do?

2a
4 marks

On the same set of axes, sketch the graphs of y = \text{sin} \; \dfrac{1}{2}\theta and y = \text{sin}(\theta + 30^{\circ}) in the interval -270^{\circ} \leq \theta \leq 270^{\circ}. Label the coordinates of points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

How did you do?

2b
2 marks

Find the solution to the equation \text{sin} \; \dfrac{1}{2}\theta = \text{sin}(\theta + 30^{\circ}) within the interval -90^{\circ} \leq \theta \leq 0^{\circ}. Hence, determine the coordinates of the corresponding point of intersection between the two graphs in part (a).

How did you do?

3a
4 marks

On the same set of axes, sketch the graphs of y = \text{tan} \; \dfrac{1}{2}\theta and y = \text{tan}(\theta - 30^{\circ}) in the interval -360^{\circ} \leq \theta \leq 360^{\circ}. Label the coordinates of points of intersection with the coordinate axes.

How did you do?

3b
3 marks

Within the interval -360^{\circ} \leq \theta \leq 360^{\circ}, determine the coordinates of the two points where \text{tan} \; \dfrac{1}{2}\theta = \text{tan}(\theta - 30^{\circ}). Give your answer in surd form.

How did you do?

4a
2 marks

The graph shows part of the curve with equation y = \text{sin}(x + k^{\circ}), where k is a constant.

Graph of part of y = sin(x + k°)

A student states that there are an infinite number of possible values for k. Is the student correct? You must explain your answer fully.

How did you do?

4b
2 marks

Another student claims that the curve could also be the graph of the equation y = \text{cos}(x + k^{\circ}). Find a value for k to show that the student is correct.

How did you do?

5a
4 marks

The graph shows two curves with equations y = p \, \text{sin} \; x and y = \text{cos}(x + q^{\circ}), in the interval -180^{\circ} \leq x \leq 180^{\circ}, where p and q are integers.

Graph of y = p sin x and y = cos(x + q°) for −180° ≤ x ≤ 180°, meeting at points R and S

Using the graph above, find the values of p and q and label or list the points of intersection each graph has with the coordinate axes.

How did you do?

5b
2 marks

Within the stated interval, the curves intersect at the two points R and S as shown in the diagram. The coordinates of point R are (9.90^{\circ},\ 0.34), accurate to 2 decimal places. By considering the graph, as well as the properties of the sine and cosine functions, state the coordinates of point S, to two decimal places.

How did you do?

6
6 marks

(i) Describe geometrically the transformation that maps the graph of y = \dfrac{1}{3} \text{tan} \; x onto the graph of y = 3 \, \text{tan} \; x.

(ii) On the graph of y = \text{tan} \; x, a point S has coordinates \left(60^{\circ},\ \sqrt{3}\right). State the new coordinates of point S after a transformation onto each of the graphs in part (i). Give your answers in surd form.

How did you do?

7a
2 marks

Describe geometrically the transformation that maps the graph of y = \text{sin}(x + 20^{\circ}) onto the graph of y = \text{cos}(x + 20^{\circ}).

How did you do?

7b
2 marks

On the same set of axes, sketch both graphs in the interval -180^{\circ} \leq x \leq 180^{\circ}. Label the coordinates of any points of intersection between the two graphs.

How did you do?

8
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6 marks

A function \text{f}(x) = \text{cos} \; px, 0^{\circ} \leq x \leq 360^{\circ}, first crosses the x-axis at 18^{\circ}.

(i) Determine the value of p and sketch the graph of y = \text{f}(x).

(ii) State the period of \text{f}(x).

How did you do?