Trigonometric Functions (Edexcel A Level Maths: Pure): Exam Questions

You are given that

By sketching an appropriate graph, find all the solutions of

in the interval .

(i) Sketch the graph of in the interval .

(ii) Use the graph to find all the values of for which in the given interval.

(i)Sketch the graph of in the interval .

The sketch must include coordinates of all points where the graph meets the coordinate axes.

(ii) Given that , use your graph to find another value of in the given range for which .

(i) On the same set of axes, sketch the graphs of and where .

Label the coordinates of all points of intersection with the coordinate axes.

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

The graph below shows the curve with equation , in the interval .

A student states that the curve could also have the equation .

Is the student correct? Give a reason for your answer.

For the graph shown in part (a), find the coordinates of all the points of intersection between the curve and the coordinate axes within the given interval.

Find a different example of an equation that represents the same curve, .

The graph below shows the curve with equation in the interval .

Point A has coordinates and is the minimum point closest to the origin.

Point B is the maximum point closest to the origin.

State the coordinates of B.

The straight line with equation meets the graph of  at the three points P, Q and R, as shown in the diagram.

Given that point P has coordinates , use the graph to find the coordinates of Q and R.

(i) Describe geometrically the transformation that maps the graph of onto the graph of .

(ii)  On the graph of , the point Q has coordinates , where is in degrees. State the coordinates of the image of the point Q on the graph of .

A section of a new rollercoaster has a series of rises and falls. The vertical displacement of the rollercoaster carriage, , measured in metres relative to a fixed reference height, can be modelled using the function

where is the time in seconds.

(i) Sketch the function for the interval .

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

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

(i) On the same set of axes, sketch the curves and  in the interval .

Label the coordinates of all points of intersection with the coordinate axes.

(ii) Find the number of solutions to the equation in the interval  .

(i) Sketch the graph of in the interval .

The sketch must include the coordinates of all points where the curve meets the coordinate axes.

(ii) Given that , use your graph to find all other values of  in the given interval for which .

You are given that .

Use the graph of in the interval to find all other values of in this interval for which

On the same set of axes, sketch the curves  and  in the interval .

Label the coordinates of all points of intersection with the coordinate axes.

In each case, state the period of the curve.

(i) Describe geometrically the transformation that maps the graph of on to the graph of .

(ii) On the graph of , the point Q has coordinates , where is in degrees. State the coordinates of the image of the point Q on the graph .

Give your answer in surd form.

(i) Describe geometrically the transformation that maps the graph of  on to the graph of . 

(ii) On the graph of , the point S has coordinates where is in degrees.

State the coordinates of point S after a transformation onto each of the graphs in part (i). Give your answers in surd form.

(i) Sketch the graph of in the interval .

(ii) Given that  , use your graph to find all values of  in the given interval for which

Show your working on the graph clearly.

Sketch the graph of in the interval  .

Use the fact that to find all the values of for which

in the interval  .

(i) On the same set of axes, sketch the curves  and  for the interval .

State the coordinates of

• any points where the curves meet the coordinate axes

• any maximum and minimum points

(ii) Show algebraically that satisfies the equation .  Hence, use your sketch to find any other solutions to the equation

in the interval  .

The graph below shows a curve with equation ,  , where is a constant.

The graph passes through the point with coordinates .

A student states that there is only one possible value for .

Explain why the student is incorrect and state at least two possible values for .

Find the coordinates of all points where the curve meets the -axis in the given interval.

The graph below shows the curve with equation in the interval .

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

State the coordinates of A and B.

  .

The straight line with equation  meets the graph  at three points, R, S, and T.

Find the coordinates of R, S, and T.

Changes in the depth of water in a small tidal estuary relative to a fixed reference depth can be modelled using the function

where  is measured in metres and is the time in hours.

(i) Sketch the function for the interval .

(ii) If represents 2pm, between which times, to the nearest half hour, will the estuary be at or above the depth of ?

A series of dips and mounds caused by underground mining has a cross-section which can be modelled using the function

where  and  are the horizontal and vertical displacements of the ground, in metres, from a fixed origin.

(i) Sketch the function for the interval and state the period of the model.

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

You are given that .

Use a suitable graph to find all the solutions to

in the interval .

(i) On the same set of axes, sketch the curves  and  in the interval .

Show clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) State the period of each function.

A function is given by , where .

The graph of first crosses the -axis at .

(i) Determine the value of  and sketch the graph of .

(ii) State the period of .

(i) On the same set of axes, sketch the curves and in the interval .

Show clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) Find the number of solutions to the equation

in the interval .

On the same set of axes, sketch the graphs of and in the interval . Label the coordinates of points of intersection with the coordinate axes and of maximum and minimum points where appropriate.

Find the solution to the equation  within the interval . Hence, determine the coordinates of the corresponding point of intersection between the two graphs in part (a).

On the same set of axes, sketch the curves and in the interval .

Show clearly the coordinates of any points where the curves meet the coordinate axes.

In the interval , find the coordinates of the two points on your sketch at which

Give your answer in surd form.

The graph below shows part of the curve with equation , where   is a constant.

A student states that there are an infinite number of possible values for .

Is the student correct? You must explain your answer.

Another student claims that the curve shown could also have the equation .

Find a value for for which this student is correct.

The graph below shows two curves with equations  and , in the interval , where  and are integers.

• The graph of passes through the point

• The graph of passes through the point

Find the values of and .

For , the curves intersect at the two points, and as shown in the diagram.

The coordinates of point  are (9.90, 0.34), to 2 decimal places.

Use these coordinates, along with diagram in part (a), to find the coordinates of point , to 2 decimal places.

Describe geometrically the transformation that maps the graph of on to the graph of .

On the same set of axes, sketch and .for the interval .

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