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When discussing trigonometric functions such as the sine, cosine, and tangent functions, it is natural to consider their reciprocal functions.
Trigonometric Function Reciprocal Trigonometric Function

This lesson will explore the graphs of these reciprocal trigonometric functions.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Communication Tower

A massive communication tower is anchored to the ground with wires.

communication tower
These wires are attached to the tower at a height of meters above the ground. The following function models the length of a wire.
Here, is the measure of the angle formed by the wire and the ground. Graph the given function to find the length of the wire that makes an angle of radians with the ground. Round the answer to the nearest integer.
Explore

Exploring the Graphs of the Cosine and Sine Functions, and Their Quotient

In the diagram, the graphs of the functions and can be seen.
sin x, cos, (cos x)/(sin x)
Consider the graph of How is it different from the other two graphs? What happens in the graph of when
Discussion

The Cotangent Function

Let be the point of intersection of the terminal side of an angle in standard position and the unit circle. The cotangent function, denoted by is defined as the ratio of the coordinate to the coordinate of

Unit Circle
Recall that the and coordinates of this point correspond to the cosine and sine of respectively. Therefore, the cotangent function can also be defined as the ratio of to

Since division by is not defined, the graph of the parent cotangent function has vertical asymptotes where This means that the graph has vertical asymptotes at every multiple of The graph of can be drawn by making a table of values.

Cotangent function y=cot(x) that has a domain (-pi,3pi) and asymptotes x=-pi, x=0, x=pi, x=2pi, and x=3pi

Consider now the general form of a cotangent function.

Here, and are non-zero real numbers and is measured in radians. The properties of the cotangent function are stated below.

Properties of
Amplitude No amplitude
Number of Cycles in
Period
Domain All real numbers except multiples of
Range All real numbers
Method

Graphing a Cotangent Function

Recall the format of a cotangent function.
Here, and are non-zero real numbers and is measured in radians. The values of and can be used to graph the function. Consider an example function.
In this function, and To sketch one cycle of a cotangent curve, its asymptotes and three points can be used. As with other trigonometric functions, there are five elements that are equally spaced through one cycle. The asymptote-point-zero-point-asymptote pattern is helpful for graphing this function. Four steps will be followed.
1
Find the Period and Cycle
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The period of a cotangent function is the quotient of and
The period of this function is This means that one cycle goes from to
2
Graph the Asymptotes
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Asymptotes occur at the end of each cycle. Therefore, the given function has asymptotes at and

asymptotes
3
Divide the Period and Locate Points
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Divide the period into fourths and locate the three equidistant points between the asymptotes. The period for this function goes from to so a table of values will be made for and

The points found in the table are and

asymptotes and points
4
Draw the Graph
expand_more

Finally, the points can be connected with a smooth curve to draw the graph for one cycle.

one cycle

Once the graph for one cycle is drawn, it can be replicated as many times as desired to draw more cycles. Here, another cycle is graphed.

two cycles

Extra

 Alternative Method
Alternatively, the graph of a cotangent function can be drawn by considering the graphs of the sine and cosine functions. To show this method, the graph of the parent cotangent function will be drawn.
To draw the graph by using this method, five steps will be followed.
1
Graph the Sine and Cosine Functions
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Start by drawing the graphs of the the sine and cosine functions on the same coordinate plane.
Sine and Cosine
2
Draw the Asymptotes
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Recall that the cotangent of an angle is the ratio of the cosine to the sine of the angle.
Therefore, the asymptotes of the cotangent function will be located where
Asymptote
3
Plot Two Zeros
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Since the cotangent of an angle is the ratio of the cosine to the sine of the angle, the cotangent function and the cosine function have the same zeros.
Therefore, the intercepts of the cotangent function occur when
Sine and Cosine
4
Plot More Points on the Curve
expand_more
By definition, the cotangent function is when the cosine and the sine have the same value. This happens at their points of intersection. Therefore, the value of the cotangent function will be at the coordinate of these points.
Sine and Cosine Cotangent
Similarly, the output of the cotangent function is when the sine and the cosine function have opposite values.
Sine and Cosine Cotangent
5
Draw the Graph
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Finally, two cycles of the cotangent function can be graphed by connecting the marked points.
Cotangent Animation
It is seen that the period of the cotangent function is Since each branch comes from positive infinity towards negative infinity, the cotangent function has no amplitude and its range is all real numbers.
Example

Tanabata Trees and a Cotangent Function

Tanabata is a Japanese festival that celebrates two mythical lovers separated by the Milky Way. A Tanabata tree is a type of tree on which people hang wishes written on paper during Tanabata. Ramsha is starting a Tanabata garden in her backyard. She realized that two of the Tanabata trees follow the path of one cycle of a cotangent function.

Error loading image.
After doing some calculations, she found the equation of the curve that these trees follow.
Draw the graph of the function for values of between and

Answer
1/2 cot(2x)

Hint

Find the period and cycle, then graph the asymptotes and plot some points. Finally, sketch the curve.

Solution
To graph a cotangent function, the first step is to calculate the period. To do so, consider the general form of a cotangent function and the given function.
Here, and The period of the function is the quotient of and
The period of the function is This means that one cycle goes from to another cycle from to and so on. Since the asymptotes occur at the end of each cycle they can be drawn on a coordinate plane.
asymptotes

Next, divide the period into fourths and locate three equidistant points between the asymptotes. Since a period goes from to a table of values will be made for and

The points found in the table are and These three points can be plotted on the plane.

points

The points are then connected with a smooth curve to graph one period of the function.

one cycle

Finally, the cycle can be replicated as many times as desired. In this case, the graph will be drawn for values of between and

1/2 cot(2x)
Explore

Exploring the Graphs of the Cosine Function and its Reciprocal Function

The graphs of the functions and can be seen in the diagram below.
sec x and cos x
What happens in the graph of at the zeros of
Discussion

The Secant Function

Let be the point of intersection of the terminal side of an angle in standard position and the unit circle. The secant function, denoted as is defined as the reciprocal of the coordinate of

Unit Circle
Recall that the coordinate of this point corresponds to the cosine of Therefore, the secant function can also be defined as the reciprocal of

Since division by is not defined, the graph of the parent secant function has vertical asymptotes where This means that the graph has vertical asymptotes at odd multiples of The graph of can be drawn by making a table of values.

The secant function y=sec(x) with asymptotes at -3pi/2, -pi/2, pi/2, 3p/2, and 5pi/2 over the domain [-3pi/2,5pi/2]

Consider the general form of a secant function.

Here, and are non-zero real numbers and is measured in radians. The properties of the secant function are be stated in the table below.

Properties of
Amplitude No amplitude
Number of Cycles in
Period
Domain All real numbers except odd multiples of
Range
Method

Graphing a Secant Function

Recall the format of a secant function.
Here, and are non-zero real numbers and is measured in radians. The values of and can be used to graph the function. Consider an example function.
In this function, and The asymptotes, some points, and the graph of the cosine function can be used to sketch one cycle of a secant curve. Then the cycle can be replicated as many times as desired. Five steps will be followed.
1
Find the Period
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The period of a secant function is the quotient of and
The period of this function is
2
Graph the Related Cosine Function
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Consider the related cosine function.
The graph of the cosine function can be drawn on a coordinate plane.
cosine function
3
Graph the Asymptotes and Plot Points
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By definition, the asymptotes of the secant function occur when the output of the cosine function is Furthermore, the maximum and minimum points of the cosine function are also points on the secant function's graph.
asymptotes
4
Divide the Period and Locate Points
expand_more

Here, the period is and the asymptotes occur every radians. Therefore, the asymptotes are located not only at the beginning and at the end of each period, but also in the middle of each period. Divide the interval between two asymptotes in fourths by the following pattern.

asymptote - point - max/min - point - asymptote

Notice that the middle point in the pattern, the maximum or minimum point, was already plotted in the previous step. The remaining four points can be found for the interval that goes from to by making a table of values.

The points from the table are and
points
5
Draw the Graph
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Finally, the points can be connected with a smooth curve to draw the graph for one cycle.

one cycle

Once the graph for one cycle is drawn, it can be replicated as many times as desired to draw more cycles. Here, one more cycle will be graphed.

two cycles
Example

A Bowl and a Secant Function

Ramsha is thinking about a sustainable way of fertilizing her new garden. She collects her kitchen scraps in a bowl so that she can compost them and use the compost for the Tanabata garden. While mixing these scraps with soil to make some fertilizer, Ramsha noticed that the shape of the bowl matches the shape of one branch of a secant function.

Error loading image.
After doing some calculations, she found the equation of the curve that models the bowl.
Draw the graph of the above function for values of between and

Answer
2 sec x

Hint

Graph the related cosine function and recall that the asymptotes of the secant function occur at the zeros of the cosine function.

Solution
To graph a secant function, the first step is to calculate the period. Consider the general form of a secant function and the given function.
Here, and The period of the function is the quotient of and
Now that it is known that the period of the function is consider the related cosine function.
This cosine function will now be graphed on a coordinate plane for values of between and
2 cos x
The asymptotes of the secant function occur at the zeros of the cosine function. Furthermore, the maximum and minimum points of the cosine function are also points on the curve of the secant function.
asymptotes
Next, more points on the curve of can be found by making a table of values.

The points found in the table can now be plotted. Finally, each set of points can be connected with smooth curves.

2 sec x

The graph of the curve of the bowl has been drawn.

Explore

Exploring the Graphs of the Sine Function and its Reciprocal Function

The graphs of the functions and are shown in the diagram below.
sec x and cos x
What happens in the graph of at the zeros of
Discussion

The Cosecant Function

Let be the point of intersection of the terminal side of an angle in standard position and the unit circle. The cosecant function, denoted as is defined as the reciprocal of the coordinate of

Unit Circle
Recall that the coordinate of this point corresponds to the sine of Therefore, the cosecant function can also be defined as the reciprocal of

Since division by is not defined, the graph of the parent cosecant function has vertical asymptotes where This means that the graph has vertical asymptotes at multiples of The graph of can be drawn by making a table of values.

cosecant function

Consider the general form of a cosecant function.

Here, and are non-zero real numbers and is measured in radians. The properties of the cosecant function are stated below.

Properties of
Amplitude No amplitude
Number of Cycles in
Period
Domain All real numbers except multiples of
Range
Method

Graphing a Cosecant Function

Recall the format of a cosecant function.
Here, and are non-zero real numbers and is measured in radians. The values of and can be used to graph the function. Consider an example function.
In this function, and To sketch one cycle of a cosecant curve, its asymptotes, some points, and the graph of the sine function can be used. Then, the cycle can be replicated as many times as desired. Five steps will be followed.
1
Find the Period
expand_more
The period of a cosecant function is the quotient of and
The period of this function is
2
Graph the Related Sine Function
expand_more
Consider the related sine function.
The graph of the sine function can be drawn on a coordinate plane.
The graph of the sine function 2*sin(1/2 theta) over the domain [-pi,7pi]
3
Graph the Asymptotes and Plot Points
expand_more
By definition, the asymptotes of the cosecant function occur when the output of the sine function is Furthermore, the maximum and minimum points of the sine function are also points on the cosecant function's graph.
asymptotes
4
Divide the Period and Locate Points
expand_more

Here, the period is and the asymptotes occur every radians. Therefore, the asymptotes are located not only at the beginning and at the end of each period, but also in the middle of each period. Divide the interval between two asymptotes in fourths by the following pattern.

asymptote - point - max/min - point - asymptote

Notice that the middle point, which is the maximum or minimum point, was already plotted in the previous step. Four more points can be found for the interval that goes from to by making a table of values.

The points found in the table are and
points
5
Draw the Graph
expand_more

Finally, each set of points can be connected with a smooth curve to draw the graph for one cycle.

one cycle

Once the graph for one cycle is drawn, it can be replicated as many times as desired to draw more cycles. Another cycle is graphed below.

two cycles
Example

An Umbrella and a Cosecant Function

To protect her new garden, Ramsha decides to set up an umbrella to cover the small plants when heavy rains are forecast. She realizes that the umbrella has the shape of one branch of a cosecant function.

Error loading image.
After doing some calculations, she found the equation of the curve that models the umbrella.
Draw the graph of the function for values of between and

Answer
1/2 csc x

Hint

Graph the related sine function and recall that the asymptotes of the cosecant function occur at the zeros of the sine function.

Solution
To graph a cosecant function, the first step is to calculate the period. Consider the general form of a cosecant function and the given function.
Here, and The period of the function is the quotient of and
The period of the function is Now consider the related sine function.
This sine function will now be graphed on a coordinate plane for values of between and
1/2 sin x
The asymptotes of the cosecant function occur at the zeros of the sine function. Furthermore, the maximum and minimum points of the sine function are also points on the curve of the cosecant function.
asymptotes
Next, more points on the curve of will be found by making a table of values.

The points found in the table can now be plotted. Finally, connect the sets of points with smooth curves.

2 sec x
Pop Quiz

Finding the Period

Find the period of the following functions. Round the answers to two decimal places.

Find the period of the function
Closure

Drawing and Using the Graph of a Cosecant Function

The challenge presented at the beginning can be solved with the topics covered in this lesson. It was given that a massive communication tower is anchored to the ground with wires.

communication tower
These wires are attached to the tower at a height of meters above the ground. The following function models the lengths of the wires.
Here, is the measure of the angle formed by the wire and the ground. Graph the function to find the length of the wire that makes an angle of radians with the ground. Round the answer to the nearest integer.

Hint

Graph the cosecant function and pay close attention to the value for

Solution
To find the length of the wire that makes an angle of radians with the ground, graph the cosecant function. The first step is to calculate the period of the function. Consider the general form of a cosecant function and the given function.
Here, and The period of the function is the quotient of and
The period of the function is Now consider the related sine function.
This sine function can be graphed on a coordinate plane.
sine function
The asymptotes of the cosecant function occur at the zeros of the sine function. Furthermore, the maximum and minimum points of the sine function are also points on the curve of the cosecant function.
max and min
Next, more points on the curve of will be found by making a table of values.

Finally, the points found in the table are be plotted and each set of points connected with smooth curves.

y = 5 csc theta

Now that the function has been graphed, the point at can be located and its coordinate identified.

y = 5 csc theta

The exact coordinate cannot be found in the graph, but its nearest integer can be identified as Therefore, the length of the wire that makes an angle of radians with the ground is about meters.


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