3.3 Trigonometric Functions

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

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

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

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.

Give the coordinates of all points of intersection with 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.

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A straight line with equation meets the graph of at three points, R, S, and T.  Determine 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, the angle is measured in radians, and t is the time in hours.

Sketch the function for the interval .

If  represents 2pm, during what times, to the nearest half hour, will the estuary be at or above the halfway point between  and its maximum depth?

A series of dips and mounds caused by underground mining has a cross-section which can be modelled using the function , where and are respectively the horizontal and vertical displacements, in metres, from a fixed origin point.

Sketch the function for the interval  and state the periodicity of the model.

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

By sketching an appropriate graph, find all the solutions of ,
in the interval  

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

in the interval 

in the interval 

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

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

A student states that the curve could also have equation .

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

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

Give another example of an equation that would also produce 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.

A straight line with equation  meets the graph of at the three points , and , as shown in the diagram.

Given that point  has coordinates  , use graph symmetries to determine the coordinates of and .

Describe geometrically the transformation that maps the graph of onto the graph of

On the graph of , a point P has coordinates .
State the new coordinates of point P after the transformation to .

Describe geometrically the transformation that maps the graph of onto the graph of

On the graph of a point has coordinates .
State the new coordinates of point after the transformation to .

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 t is the time in seconds.

Sketch the function for the interval .

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

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

Deduce 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 graphs of  and  in the interval . Label the coordinates of points of intersection with the coordinate axes.

Within the interval , determine the coordinates of the two points where .  Give your answer in surd form.

The graph below shows part of the curve with equation ,  where the angle is measured in radians and k is a constant.

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

Another student claims that the curve could also be the graph of the equation . Find a value for k to show that the student is correct.

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

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

Within the stated interval, the curves intersect at the two points and  as shown in the diagram. The coordinates of point 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.

Describe geometrically the transformation that maps the graph of onto the graph of  .

On the same set of axes, sketch both graphs in the interval .

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

The graph below shows the curve with equation , in the interval . One value of  has been labelled .

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

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A function ,  first crosses the -axis at .

Determine the value of  and sketch the graph of .

State the period of .