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International ASMaths: Pure 1EdexcelExam Questions3. Trigonometry3.3 Trigonometric Functions

Trigonometric Functions (Edexcel International AS Maths: Pure 1)

Exam Questions

3 hours43 questions
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1
Sme Calculator6 marks

On separate diagrams sketch the graphs of:

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

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

y equals tan space x           negative 180 degree less or equal than x less or equal than 180 degree 

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2
Sme Calculator3 marks

Sketch the graph of  for y equals sin space 2 x spacefor 0 less or equal than x less or equal than pi.

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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.

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4
Sme Calculator3 marks

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

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

(i)
y equals f left parenthesis x right parenthesis plus 2
(ii)
y equals f left parenthesis 3 x right parenthesis
(iii)
y equals f left parenthesis x plus 30 degree right parenthesis

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5
Sme Calculator2 marks

Write down the values for which cos space x space equals space 1 half, for 0 less or equal than x less or equal than 2 pi.

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6
Sme Calculator2 marks

The diagram below shows the graph ofy space equals space tan space x commafor 0 degree space less or equal than space x space less or equal than space 720 degree.

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

screenshot-2023-07-20-at-9-44-58-pm

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7
Sme Calculator2 marks

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8
Sme Calculator3 marks

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

(i)
2 straight f left parenthesis theta right parenthesis plus 3
(ii)
3 straight f left parenthesis 2 theta right parenthesis

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9
Sme Calculator2 marks

Write down all the values of x for which sin space 3 x equals 0, where 0 less or equal than x less or equal than 2 pi. 

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1
Sme Calculator3 marks
(i)
Sketch the graph of y space equals space cos 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)
Write down all the values of theta for which cos space theta space equals space 0 spacefor 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 space equals space sin space theta in the interval 0 less or equal than theta less or equal than 2 pi. The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii)     Given that sin straight pi over 6 equals 0.5, use your graph to find another value of theta in the given

          range for which sin space theta space equals space 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 space equals space cos space open parentheses theta space plus space 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 space left parenthesis theta plus 30 degree right parenthesis space equals space 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 space sin space theta in the interval 0 less or equal than theta less or equal than 2 pi.

(ii)
y equals negative 2 space sin theta in the interval 0 less or equal than theta less or equal than 2 pi.

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 theta and y equals cos space 3 theta 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 left parenthesis x plus 50 degree right parenthesis, in the interval negative 180 degree less or equal than x less or equal than 360 degree

screenshot-2023-07-21-at-1-56-37-pm

A student states that the curve could also have equation y equals tan left parenthesis x minus 130 degree right parenthesis.

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

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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 interval negative straight pi over 3 less or equal than x less or equal than fraction numerator 3 straight pi over denominator 2 end fraction.

screenshot-2023-07-21-at-2-08-05-pm

Point A has coordinates open parentheses negative straight pi over 4 comma space minus 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 space equals space minus 1 half meets the graph of y space equals space sin space 2 x at the three points P, Q and R, as shown in the diagram.

Given that point P has coordinates open parentheses negative straight pi over 12 comma space minus 1 half close parentheses , use graph symmetries to determine the coordinates of Q and R.

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9
Sme Calculator3 marks

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

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

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10
Sme Calculator3 marks

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

On the graph of y space equals space sin space x comma spacea point Q has coordinates open parentheses straight pi over 6 comma space 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 space equals space 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 function y equals 30 space cos left parenthesis 24 t right parenthesis degree comma  where t is the time in seconds.

(i)
Sketch the function for the interval 0 less or equal than 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

On the same set of axes, sketch the graphs of y space equals space sin space 2 theta and y space equals space cos space open parentheses theta space plus space straight pi over 2 close parentheses in the interval negative straight pi space less or equal than straight theta less or equal than straight pi.

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

Deduce the number of solutions to the equation sin space 2 theta space equals space cos space open parentheses theta plus straight pi over 2 close parentheses in the interval negative straight pi less or equal than straight theta less or equal than straight pi.

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1
Sme Calculator2 marks
(i)
Sketch the graph of y equals space cos space theta in the interval negative straight pi over 2 less or equal than space theta space less or equal than 2 straight pi. The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii)
Given that cos space straight pi over 3 space equals space 0.5 comma use your graph to find all other values of theta in the given interval for which cos space theta space equals space 0.5.



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2
Sme Calculator3 marks
(i)
Sketch the graph of y space equals tan space theta in the interval negative space 270 degree space less or equal than space theta space less or equal than 270 degree. The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii)
Given that tan space 30 degree space equals space 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 space equals space 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 space equals space sin space theta in the interval negative 180 degree space 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 that sin space 60 degree space equals space fraction numerator square root of 3 over denominator 2 end fraction, use your graph to find all values of theta in the given interval for which sin space theta space equals space minus fraction numerator square root of 3 over denominator 2 end fraction.

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4
Sme Calculator4 marks
(i)
Sketch the graph of y space equals space tan space open parentheses theta minus straight pi over 4 close parentheses space  in the interval negative 2 straight pi space less or equal than space straight theta space less or equal than 2 straight pi.
(ii)
Write down all the values of theta for which tan space open parentheses theta space minus space straight pi over 4 close parentheses space equals space 1 in the given interval.

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5
Sme Calculator6 marks

On the same set of axes, sketch the graphs of y space equals space minus 3 cos space theta and y space equals space cos space 3 theta in the interval 0 space less or equal than space theta space less or equal than space 2 straight pi.

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 of y space equals space 1 half sin space theta and y space equals space sin space open parentheses theta space minus space 60 degree close parentheses in the interval negative space 180 degree space less or equal than space theta space less or equal than space 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 space equals space 90 degree is a solution to the equation 1 half sin space theta space equals space sin space open parentheses theta space minus space 60 degree close parentheses.  Hence, using your sketch from part (i) or otherwise, find all other solutions to the equation 1 half sin space theta space equals space sin open parentheses theta space minus space 60 degree close parentheses in the interval negative 180 degree space less or equal than space theta space less or equal than space 180 degree.

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7a
Sme Calculator2 marks

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

screenshot-2023-07-22-at-11-34-50-am

 

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 the curve with equation y space equals space sin space 3 x in the interval negative straight pi over 3 space less or equal than space x space less or equal than space straight pi.

screenshot-2023-07-22-at-11-41-12-am

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


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8b
Sme Calculator2 marks

A straight line with equation y space equals space fraction numerator square root of 3 over denominator 2 end fraction meets the graph of y space equals space sin space 3 x 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 space tan space x onto the graph of y space equals space 1 fifth space tan space x.
(ii)
On the graph of y space equals space 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 space equals space 1 fifth space 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 space equals space open parentheses sin space straight pi over 8 t close parentheses,  where y is measured in metres, the angle is measured in radians, and t is the time in hours.

(i)
Sketch the function for the interval 0 space less or equal than space t space less or equal than space 8.

(ii)
If t space equals space 0 represents 2pm, during what times, to the nearest half hour, will the estuary be at or above the halfway point between y space equals space 0 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 space equals space 4 cos space open parentheses 18 x close parentheses degree, 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 space less or equal than space x space less or equal than space 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 space equals space tan space 1 fourth theta and y space equals space cos space open parentheses theta space plus space 120 degree close parentheses in the interval 0 degree space less or equal than space theta space less or equal than space 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 space open parentheses theta space plus space 120 degree close parentheses space minus space tan space 1 fourth space theta space equals space 0, in the interval 0 degree space less or equal than space theta space less or equal than space 270 degree space.

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1
Sme Calculator4 marks

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

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2
Sme Calculator6 marks
(i)
On the same set of axes, sketch the graphs of y space equals space cos open parentheses negative 2 theta close parentheses  and y space equals space cos space 1 half theta in the interval negative 2 straight pi space less or equal than space straight theta space less or equal than space 2 straight pi.  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 space equals space sin space 1 half theta and y equals space sin space open parentheses theta plus 30 degree close parentheses in the interval negative space 270 degree space less or equal than space theta space less or equal than space 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 space 1 half space theta space equals space sin space open parentheses theta space plus space 30 degree close parentheses within the interval negative 90 degree space less or equal than space theta less or equal than space 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 space equals space tan space 1 half theta and y space equals space tan space open parentheses theta space minus space straight pi over 6 close parentheses in the interval negative 2 straight pi space less or equal than space straight theta space less or equal than space 2 straight pi. Label the coordinates of points of intersection with the coordinate axes.

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4b
Sme Calculator3 marks

Within the interval negative 2 straight pi space less or equal than space straight theta space less or equal than space 2 straight pi, determine the coordinates of the two points where tan space 1 half theta space equals space tan space open parentheses theta space minus space straight pi over 6 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 space equals space sin space open parentheses x space plus space k close parentheses,  where the angle is measured in radians and k is a constant.

screenshot-2023-07-22-at-8-00-27-pm

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 space equals space cos space open parentheses x space plus space k 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 space equals space p space sin space x and y space equals space cos open parentheses space x space plus space q degree close parentheses, in the interval negative space 180 degree space less or equal than space x space less or equal than space 180 degree, where p and q are integers.

screenshot-2023-07-22-at-8-15-18-pm

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 R are (9.90°, 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 space equals space 1 third space tan space x onto the graph of y space equals space 3 space tan space x.

(ii)
On the graph of y space equals space tan space x, a point S has coordinates open parentheses straight pi over 3 comma space 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 space equals space sin space open parentheses x space plus space 20 degree close parentheses onto the graph of  y space equals space cos open parentheses space x space plus space 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 space 180 degree space less or equal than space x space less or equal than space 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 space equals space 2 space sin space theta, in the interval 0 degree space less or equal than space theta space less or equal than space 360 degree. One value of theta has been labelled open parentheses theta space equals space a degree close parentheses.

screenshot-2023-07-22-at-8-35-28-pm

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

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

 

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10
Sme Calculator6 marks

A function straight f open parentheses straight x close parentheses space equals space cos space straight p x comma space0 space less or equal than space x space less or equal than space 2 straight pi,  first crosses the x-axis at straight pi over 10.

(i)
Determine the value of p and sketch the graph of y space equals space straight f open parentheses straight x close parentheses.

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

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