ASMaths: PureOCRExam Questions5. Trigonometry5.2 Trigonometric Functions

Trigonometric Functions (OCR AS Maths: Pure)

Exam Questions

3 hours43 questions
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Very Hard
1
Sme Calculator6 marks

On separate diagrams sketch the graphs of:

(i)

y equals sin space x Ā  Ā  Ā  Ā negative 180 degree less or equal than space x less or equal than 180 degree

(ii)

y equals cos space xĀ  Ā  Ā  Ā  Ā  0 degree less or equal than space x less or equal than 360 degree

(iii)
y equals tan space xĀ  Ā  Ā  Ā  negative 180 degree less or equal than space x space less or equal than 180 degree

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2
Sme Calculator3 marks

Sketch the graph of y equals sin space 2 x forĀ 0 degree less or equal than space x less or equal than 180 degree .

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3
Sme Calculator2 marks
(i)
Write down the maximum value of y where y equals 3 space cos space x.

(ii)
Write down the minimum value of y where y equals 9 space sin space x.Ā 

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4
Sme Calculator3 marks

The point PĀ has coordinates (90Ā° , 1) and lies on the graph of y equals sin space x, where 0 degree less or equal than space x less or equal than 180 degree

Write down the coordinates of the image of point Ā under the following transformations:

(i)

y equals straight f open parentheses x close parentheses plus 2

(ii)

y equals straight f open parentheses 3 x close parentheses

(iii)
y equals straight f open parentheses x plus 30 degree close parentheses

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5
Sme Calculator2 marks

Write down the values for which cos space x equals 1 half, for 0 degree less or equal than space x less or equal than 360 degree.

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6
Sme Calculator2 marks

The diagram below shows the graph of y equals tan space x, for 0 degree less or equal than space x less or equal than 720 degree .

By adding a suitable line to the graph, show that there are four solutions to the equation tan space x equals 2, for 0 degree less or equal than space x less or equal than 720 degree.

q6-5-2-trigonometry-functions-edexcel-a-level-pure-maths-easy

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7
Sme Calculator2 marks

Sketch the graph of y equals negative sin space straight thetaĀ for 0 degree less or equal than straight theta less or equal than 360 degree.

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8
Sme Calculator3 marks

Given that straight f open parentheses theta close parentheses equals cos space theta, write the following functions in terms of cos space theta.

(i)

2 straight f open parentheses theta close parentheses plus 3

(ii)
3 straight f open parentheses 2 theta close parentheses

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9
Sme Calculator2 marks

Write down all the values of x for whichĀ sin space 3 x degree equals space 0, where 0 degree less or equal than space x less or equal than space 360 degree.

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1
Sme Calculator3 marks
(i)

Sketch the graph of y equals cos space theta in the interval 0 degree less or equal than theta space less or equal than 360 degree. The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii)
Write down all the values of theta for which cos space theta equals 0 forĀ 0 degree less or equal than theta less or equal than 360 degree .

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2
Sme Calculator3 marks
(i)

Sketch the graph of y equals sin space theta in the interval 0 degree less or equal than theta less or equal than 360 degree. The sketch must include coordinates of all points where the graph meets the coordinate axes

(ii)
Given that sin space 30 degree equals 0.5, use your graph to find another value of theta in the given range for which sin space theta equals 0.5.

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3
Sme Calculator3 marks

By sketching an appropriate graph, find all the solutions of tan space theta equals negative 1,
in the interval 0 degree less or equal than theta less or equal than 360 degree.

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4
Sme Calculator4 marks
(i)

Sketch the graph of y equals cos open parentheses theta plus 30 degree close parentheses in the interval negative 180 degree less or equal than theta less or equal than 360 degree.

(ii)
Write down all the values where cos open parentheses theta plus 30 degree close parentheses equals 0 in the given interval.

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5
Sme Calculator5 marks

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

(i)

y equals 2 sin space theta in the interval 0 degree less or equal than theta less or equal than 360 degree

(ii)
y equals negative 2 sin space theta in the interval Ā 0 degree less or equal than space theta less or equal than 360 degree

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

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6
Sme Calculator5 marks
(i)

On the same set of axes, sketch the graphs of y equals cos space thetaand y equals cos space 3 thetaĀ in the interval negative 180 degree less or equal than theta less or equal than 180 degree, giving the coordinates of all points of intersection with the coordinate axes.

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

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7a
Sme Calculator2 marks

The graph below shows the curve with equation y equals tan open parentheses x plus 50 degree close parentheses, in the interval negative 180 degree less or equal than space x less or equal than 360 degree

q7a-5-2-trigonometry-functions-edexcel-a-level-pure-maths-medium

A student states that the curve could also have equation y equals tan open parentheses x minus 130 degree close parentheses.

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


Ā  .

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7b
Sme Calculator2 marks

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

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7c
Sme Calculator1 mark

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

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8a
Sme Calculator1 mark

The graph below shows the curve with equation y equals sin space 2 x in the intervalĀ negative 60 degree less or equal than space x less or equal than 270 degree

q8a-5-2-trigonometry-functions-edexcel-a-level-pure-maths-medium.Point A has coordinates open parentheses negative 45 degree comma negative 1 close parentheses and is the minimum point closest to the origin. Point B is the maximum point closest to the origin. State the coordinates of B.

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8b
Sme Calculator2 marks

A straight line with equation y equals negative begin inline style 1 half end style meets the graph of y equals sin space 2 xĀ at the three points P, Q and R, as shown in the diagram.

Given that point P has coordinates open parentheses negative 15 degree comma negative begin inline style 1 half end style close parentheses , use graph symmetries to determine the coordinates of Q and R.

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9
Sme Calculator3 marks
(i)

Describe geometrically the transformation that maps the graph of y equals cos space x onto the graph of y equals 4 space cos space x.

(ii)
On the graph of y equals cos space x, a point P has coordinates open parentheses 60 degree comma 0.5 close parentheses.
State the new coordinates of point P after the transformation to y equals 4 space cos space x.

Ā 

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10
Sme Calculator3 marks
(i)

Describe geometrically the transformation that maps the graph of y equals sin space xonto the graph of y equals sin space 3 x.

(ii)Ā  Ā  Ā  Ā On the graph of y equals sin space x, a point Q has coordinates open parentheses 30 degree comma fraction numerator square root of 3 over denominator 2 end fraction close parentheses.

Ā  Ā  Ā  Ā  Ā  Ā  State the new coordinates of point Q after the transformation toĀ y equals sin space 3 x

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11
Sme Calculator6 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 functionĀ y equals 30 space cos open parentheses 24 straight t close parentheses degree,Ā where t is the time in seconds.

(i)

Sketch the function for the interval 0 less or equal than straight t less or equal than 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?

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12
Sme Calculator8 marks
(i)
On the same set of axes, sketch the graphs of y equals sin space 2 theta and y equals cos open parentheses theta plus 90 degree close parenthesesĀ in the interval negative 180 degree less or equal than theta less or equal than 180 degree.

Show clearly the coordinates of all points of intersection with the coordinate axes.

(ii)
Deduce the number of solutions to the equation sin space 2 theta equals cos open parentheses theta plus 90 degree close parentheses in the intervalĀ  negative 180 degree less or equal than theta less or equal than 180 degree.

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1
Sme Calculator2 marks
(i)

Sketch the graph of y equals cos space theta in the interval negative 90 degree less or equal than theta less or equal than 360 degree. The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii)
Given that cos space 60 degree equals 0.5, use your graph to find all other values of thetaĀ in the given interval for which cos space theta equals 0.5.

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2
Sme Calculator3 marks

Sketch the graph of y equals tan space theta in the interval negative 270 degree less or equal than theta less or equal than 270 degree. The sketch must include coordinates of all points where the graph meets the coordinate axes.

Given that tan space 30 degree equals fraction numerator 1 over denominator square root of 3 end fraction, use your graph to find all other values of theta in the given interval for which tan space theta equals fraction numerator 1 over denominator square root of 3 end fraction.

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3
Sme Calculator3 marks
(i)

Sketch the graph of y equals sin space theta in the interval negative 180 degree less or equal than theta less or equal than 180 degree. The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii)
Given thatĀ  sin space 60 degree equals fraction numerator square root of 3 over denominator 2 end fraction, use your graph to find all values of thetaĀ in the given interval for which Ā sin space theta equals negative begin inline style fraction numerator square root of 3 over denominator 2 end fraction end style.

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4
Sme Calculator4 marks

Sketch the graph of y equals tan open parentheses theta minus 45 degree close parentheses in the intervalĀ Ā negative 360 degree less or equal than theta less or equal than 360 degree.

Write down all the values of theta for which tan open parentheses theta minus 45 degree close parentheses equals 1 inĀ theĀ given interval.

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5
Sme Calculator6 marks

On the same set of axes, sketch the graphs of y equals negative 3 space cos space thetaĀ and y equals cos space 3 thetaĀ in the interval 0 degree less or equal than theta less or equal than 360 degree.

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

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6
Sme Calculator8 marks
(i)

On the same set of axes, sketch the graphs ofĀ  y equals 1 half sin space theta and y equals sin space open parentheses theta minus 60 degree close parenthesesĀ in the interval negative 180 degree less or equal than theta less or equal than 180 degree. State the coordinates of all points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

(ii)
Verify that theta equals 90 degree is a solution to the equation begin inline style 1 half end style sin space theta equals sin open parentheses theta minus 60 degree close parentheses.Ā  Hence, using your sketch from part (i) or otherwise, find all other solutions to the equation begin inline style 1 half end style sin space theta equals sin open parentheses theta minus 60 degree close parentheses in the intervalĀ negative 180 degree less or equal than theta less or equal than 180 degree .

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7a
Sme Calculator2 marks

The graph below shows a curve with equation y equals cos open parentheses x plus k degree close parentheses, negative 360 degree less or equal than x less or equal than 360 degreeĀ , where k is a constant.

q7a-5-2-trigonometry-functions-edexcel-a-level-pure-maths-hard

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.

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7b
Sme Calculator2 marks

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

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8a
Sme Calculator1 mark

The graph below shows a curve with equation y equals sin space 3 x in the intervalĀ negative 60 degree less or equal than space x less or equal than 150 degree

q8a-5-2-trigonometry-functions-edexcel-a-level-pure-maths-hard

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

Ā 
.

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8b
Sme Calculator2 marks

A straight line with equationĀ y equals begin inline style fraction numerator square root of 3 over denominator 2 end fraction end style meets the graph y equals sin space 3 xĀ at three points, R, S, and T. Determine the coordinates of R, S, and T.

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9
Sme Calculator3 marks
(i)

Describe geometrically the transformation that maps the graph of y equals tan space x onto the graph of y equals 1 fifth tan space x.

(ii)
On the graph of y equals tan space x, a point Q has coordinates open parentheses 30 degree comma fraction numerator square root of 3 over denominator 2 end fraction close parentheses.
State the new coordinates of point Q after the transformation to y equals 1 fifth tan space x.
Leave your answer in surd form.

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10
Sme Calculator5 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 equals sin open parentheses 22.5 t close parentheses degree, where yĀ is measured in metres and t is the time in hours.

(i)

Sketch the function for the interval 0 less or equal than t less or equal than 8.

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

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11
Sme Calculator5 marks

A series of dips and mounds caused by underground mining has a cross-section which can be modelled using the function y equals 4 space cos open parentheses 18 x close parentheses degree, where xĀ and yĀ are respectively the horizontal and vertical displacements, in metres, from a fixed origin point.

(i)

Sketch the function for the interval 0 less or equal than space x less or equal than 40 and state the periodicity of the model.

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

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12
Sme Calculator8 marks
(i)

On the same set of axes, sketch the graphs of y equals tan 1 fourth theta and y equals cos open parentheses theta plus 120 degree close parentheses in the interval 0 degree less or equal than theta less or equal than 270 degree. Show clearly the coordinates of all points of intersection with the coordinate axes.

(ii)
Deduce the number of solutions to the equation cos open parentheses theta plus 120 degree close parentheses minus tan 1 fourth theta equals 0, in the interval 0 degree less or equal than theta less or equal than 270 degree.

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1
Sme Calculator4 marks

By sketching an appropriate graph, find all the solutions to tan space theta equals fraction numerator negative 1 over denominator square root of 3 end fraction, in the intervalĀ 0 degree less or equal than theta less or equal than 360 degree.

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2
Sme Calculator6 marks
(i)
On the same set of axes, sketch the graphs of y equals cos open parentheses negative 2 theta close parenthesesĀ and y equals cos 1 half thetaĀ in the interval negative 360 degree less or equal than theta less or equal than 360 degree. Label the axes appropriately to show all points of intersection between the graphs and the coordinate axes.

(ii)
State the periodicity of each function.

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3a
Sme Calculator4 marks

On the same set of axes, sketch the graphs of y equals sin 1 half theta and y equals sin open parentheses theta plus 30 degree close parentheses in the interval negative 270 degree less or equal than theta less or equal than 270 degree. Label the coordinates of points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

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3b
Sme Calculator2 marks

Find the solution to the equation sin 1 half theta equals sin open parentheses theta plus 30 degree close parenthesesĀ within the intervalĀ negative 90 degree less or equal than theta less or equal than 0 degree. Hence, determine the coordinates of the corresponding point of intersection between the two graphs in part (a).

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4a
Sme Calculator4 marks

On the same set of axes, sketch the graphs of y equals tan 1 half theta and y equals tan open parentheses theta minus 30 degree close parentheses in the intervalĀ negative 360 degree less or equal than theta less or equal than 360 degree. Label the coordinates of points of intersection with the coordinate axes.

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4b
Sme Calculator3 marks

Within the interval negative 360 degree less or equal than theta less or equal than 360 degree, determine the coordinates of the two points where tan 1 half theta equals tan open parentheses theta minus 30 degree close parentheses. Give your answer in surd form.

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5a
Sme Calculator2 marks

The graph below shows part of the curve with equation y equals sin open parentheses x plus k degree close parentheses, where kĀ  is a constant.

q5a-5-2-trigonometry-functions-edexcel-a-level-pure-maths-veryhard

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

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5b
Sme Calculator2 marks

Another student claims that the curve could also be the graph of the equation y equals cos open parentheses x plus k degree close parentheses. Find a value for k to show that the student is correct.

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6a
Sme Calculator4 marks

The graph below shows two curves with equations y equals p space sin space xĀ and y equals cos open parentheses x plus q degree close parentheses, in the interval negative 180 degree less or equal than space x less or equal than 180 degree, where pĀ and q are integers.

q6a-5-2-trigonometry-functions-edexcel-a-level-pure-maths-veryhard

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

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6b
Sme Calculator2 marks

Within the stated interval, the curves intersect at the two points R and S as shown in the diagram. The coordinates of point RĀ are (9.90Ā°, 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.

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7
Sme Calculator6 marks
(i)

Describe geometrically the transformation that maps the graph of y equals begin inline style 1 third end style tan space xĀ onto the graph of y equals 3 space tan space x.Ā 

(ii)
On the graph of y equals space tan space x, a point S has coordinates open parentheses 60 degree comma square root of 3 close parentheses. State the new coordinates of point S after a transformation onto each of the graphs in part (i). Give your answers in surd form.

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8a
Sme Calculator2 marks

Describe geometrically the transformation that maps the graph of y equals sin open parentheses x plus 20 degree close parentheses onto the graph of y equals cos open parentheses x plus 20 degree close parentheses.

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8b
Sme Calculator2 marks

On the same set of axes, sketch both graphs in the interval negative 180 degree less or equal than space x less or equal than 180 degree.

Label the coordinates of any points of intersection between the two graphs.

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9
Sme Calculator2 marks

The graph below shows the curve with equation y equals 2 space sin space theta, in the intervalĀ  0 degree less or equal than theta less or equal than 360 degree. One value of thetaĀ has been labelled open parentheses theta equals a degree close parentheses.

q9-5-2-trigonometry-functions-edexcel-a-level-pure-maths-veryhardĀ 

Use the graph, along with the symmetry properties of the sine function, to verify thatĀ 

Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  Ā  2 space sin space a equals 2 space sin open parentheses 180 degree minus a close parentheses equals negative 2 space sin open parentheses 180 degree plus a close parentheses equals negative 2 space sin open parentheses 360 degree minus a close parentheses.

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10
Sme Calculator6 marks

A function straight f open parentheses x close parentheses equals cos space p x,0 degree less or equal than space x less or equal than 360 degree, first crosses the x-axis at 18 degree.

(i)

Determine the value of pĀ and sketch the graph of y equals straight f open parentheses x close parentheses.

(ii)
State the period of straight f open parentheses x close parentheses.

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