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

### Catch-Up and Review

Trigonometric Function | Reciprocal Trigonometric Function |
---|---|

$y=sinx$ | $y=sinx1 $ |

$y=cosx$ | $y=cosx1 $ |

$y=tanx$ | $y=tanx1 $ |

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

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

Challenge

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

These wires are attached to the tower at a height of $5$ meters above the ground. The following function models the length of a wire.$y=5cscθ $

Here, $θ$ 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 $4π $ radians with the ground. Round the answer to the nearest integer. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"meters","answer":{"text":["7"]}}

Explore

In the diagram, the graphs of the functions $y=sinx,$ $y=cosx,$ and $y=sinxcosx $ can be seen.

Consider the graph of $y=sinxcosx .$ How is it different from the other two graphs? What happens in the graph of $y=sinxcosx $ when $sinx=0?$

Discussion

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

Recall that the $x-$ and $y-$coordinates of this point correspond to the cosine and sine of $θ,$ respectively. Therefore, the cotangent function can also be defined as the ratio of $cosθ$ to $sinθ.$

$cotθ=sinθcosθ $

Since division by $0$ is not defined, the graph of the parent cotangent function $y=cotx$ has vertical asymptotes where $sinx=0.$ This means that the graph has vertical asymptotes at every multiple of $π.$ The graph of $y=cotx$ can be drawn by making a table of values.

Consider now the general form of a cotangent function.

$y=acotbx$

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

Properties of $y=acotbx$ | |
---|---|

Amplitude | No amplitude |

Number of Cycles in $[0,2π]$ | $2∣b∣$ |

Period | $∣b∣π $ |

Domain | All real numbers except multiples of $∣b∣π $ |

Range | All real numbers |

Method

Recall the format of a cotangent function.
*asymptote-point-zero-point-asymptote* pattern is helpful for graphing this function. Four steps will be followed.
*expand_more*
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*expand_more*
### Extra

Alternative Method

$y=acotbθ $

Here, $a$ and $b$ are non-zero real numbers and $θ$ is measured in radians. The values of $a$ and $b$ can be used to graph the function. Consider an example function.
$y=3cot21 θ $

In this function, $a=3$ and $b=21 .$ 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 1

Find the Period and Cycle

The period of a cotangent function is the quotient of $π$ and $∣b∣.$

$Period∣b∣π ⇒∣∣∣ 21 ∣∣∣ π =2π $

The period of this function is $2π.$ This means that one cycle goes from $0$ to $2π.$ 2

Graph the Asymptotes

Asymptotes occur at the end of each cycle. Therefore, the given function has asymptotes at $θ=0$ and $θ=2π.$

3

Divide the Period and Locate Points

Divide the period into fourths and locate the three equidistant points between the asymptotes. The period for this function goes from $0$ to $2π,$ so a table of values will be made for $θ=2π ,$ $θ=π,$ and $θ=23π .$

$θ$ | $3cot(21 θ)$ | $y$ |
---|---|---|

$2π $ | $3cot(21 ⋅2π )$ | $3$ |

$π$ | $3cot(21 ⋅π)$ | $0$ |

$23π $ | $3cot(21 ⋅23π )$ | $-3$ |

The points found in the table are $(2π ,3),$ $(π,0),$ and $(23π ,-3).$

4

Draw the Graph

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

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.
*expand_more*
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*expand_more*
*expand_more*

$y=cotx $

To draw the graph by using this method, five steps will be followed.
1

Graph the Sine and Cosine Functions

Start by drawing the graphs of the the sine and cosine functions on the same coordinate plane.

2

Draw the Asymptotes

3

Plot Two Zeros

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.

$cotθ=0⇔cosθ=0 $

Therefore, the $x-$intercepts of the cotangent function occur when $cosx=0.$
4

Plot More Points on the Curve

By definition, the cotangent function is $1$ when the cosine and the sine have the same value. This happens at their points of intersection. Therefore, the $y-$value of the cotangent function will be $1$ at the $x-$coordinate of these points.

Similarly, the output of the cotangent function is $-1$ when the sine and the cosine function have opposite values.

5

Draw the Graph

Finally, two cycles of the cotangent function can be graphed by connecting the marked points.

It is seen that the period of the cotangent function is $π.$ 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 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.
After doing some calculations, she found the equation of the curve that these trees follow.
### Answer

### Hint

### Solution

External credits: Kumiko

$y=21 cot2x $

Draw the graph of the function for values of $x$ between $0$ and $2π.$
Find the period and cycle, then graph the asymptotes and plot some points. Finally, sketch the curve.

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.

$General Form:Given Function: y=acotbxy=21 cot2x $

Here, $a=21 $ and $b=2.$ The period of the function is the quotient of $π$ and $∣b∣.$
$Period ∣b∣π ⇒2π $

The period of the function is $2π .$ This means that one cycle goes from $0$ to $2π ,$ another cycle from $2π $ to $π,$ and so on. Since the asymptotes occur at the end of each cycle they can be drawn on a coordinate plane.
Next, divide the period into fourths and locate three equidistant points between the asymptotes. Since a period goes from $0$ to $2π ,$ a table of values will be made for $x=8π ,$ $x=4π ,$ and $x=83π .$

$x$ | $21 cot2x$ | $y$ |
---|---|---|

$8π $ | $21 cot2(8π )$ | $21 $ |

$4π $ | $21 cot2(4π )$ | $0$ |

$83π $ | $21 cot2(83π )$ | $-21 $ |

The points found in the table are $(8π ,21 ),$ $(4π ,0),$ and $(83π ,-21 ).$ These three points can be plotted on the plane.

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

Finally, the cycle can be replicated as many times as desired. In this case, the graph will be drawn for values of $x$ between $0$ and $2π.$

Explore

Discussion

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

Recall that the $x-$coordinate of this point corresponds to the cosine of $θ.$ Therefore, the secant function can also be defined as the reciprocal of $cosθ.$

$secθ=cosθ1 $

Since division by $0$ is not defined, the graph of the parent secant function $y=secx$ has vertical asymptotes where $cosx=0.$ This means that the graph has vertical asymptotes at odd multiples of $x=2π .$ The graph of $y=secx$ can be drawn by making a table of values.

Consider the general form of a secant function.

$y=asecbx$

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

Properties of $y=asecbx$ | |
---|---|

Amplitude | No amplitude |

Number of Cycles in $[0,2π]$ | $∣b∣$ |

Period | $∣b∣2π $ |

Domain | All real numbers except odd multiples of $2∣b∣π $ |

Range | $(-∞,-∣a∣]∪[∣a∣,∞)$ |

Method

Recall the format of a secant function.
*expand_more*
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*expand_more*

The points from the table are $(-2π ,2.83),$ $(2π ,2.83),$ $(23π ,-2.83),$ and $(25π ,-2.83).$
*expand_more*

$y=asecbθ $

Here, $a$ and $b$ are non-zero real numbers and $θ$ is measured in radians. The values of $a$ and $b$ can be used to graph the function. Consider an example function.
$y=2sec21 θ $

In this function, $a=2$ and $b=21 .$ 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

The period of a secant function is the quotient of $2π$ and $∣b∣.$

$Period∣b∣2π ⇒∣∣∣ 21 ∣∣∣ 2π =4π $

The period of this function is $4π.$ 2

Graph the Related Cosine Function

Consider the related cosine function.

$SecantFunctiony=2sec21 θ Related CosineFunctiony=2cos21 θ $

The graph of the cosine function can be drawn on a coordinate plane. 3

Graph the Asymptotes and Plot Points

By definition, the asymptotes of the secant function occur when the output of the cosine function is $0.$ Furthermore, the maximum and minimum points of the cosine function are also points on the secant function's graph.

4

Divide the Period and Locate Points

Here, the period is $4π$ and the asymptotes occur every $2π$ 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.

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 $-π$ to $3π$ by making a table of values.

$θ$ | $2sec21 θ$ | $y$ |
---|---|---|

$-2π $ | $2sec(21 (-2π ))$ | $22 ≈2.83$ |

$2π $ | $2sec(21 ⋅2π )$ | $22 ≈2.83$ |

$23π $ | $2sec(21 ⋅23π )$ | $-22 ≈-2.83$ |

$25π $ | $2sec(21 ⋅25π )$ | $-22 ≈-2.83$ |

5

Draw the Graph

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

Example

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.
After doing some calculations, she found the equation of the curve that models the bowl.
### Answer

### Hint

### Solution

External credits: Kumiko

$y=2secx $

Draw the graph of the above function for values of $x$ between $-23π $ and $25π .$
Graph the related cosine function and recall that the asymptotes of the secant function occur at the zeros of the cosine function.

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.

$General Formy=asecbxGiven Functiony=2secx⇔y=2sec1x $

Here, $a=2$ and $b=1.$ The period of the function is the quotient of $2π$ and $∣b∣.$
$Period∣b∣2π ⇒∣1∣2π =2π $

Now that it is known that the period of the function is $2π,$ consider the related cosine function.
$SecantFunctiony=2secx Related CosineFunctiony=2cosx $

This cosine function will now be graphed on a coordinate plane for values of $x$ between $-23π $ and $25π .$
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.
Next, more points on the curve of $y=2secx$ can be found by making a table of values.

$x$ | $2secx$ | $y$ |
---|---|---|

$-45π $ | $2sec(-45π )$ | $-22 ≈-2.83$ |

$-43π $ | $2sec(-43π )$ | $-22 ≈-2.83$ |

$-4π $ | $2sec(-4π )$ | $22 ≈2.83$ |

$4π $ | $2sec4π $ | $22 ≈2.83$ |

$43π $ | $2sec43π $ | $-22 ≈-2.83$ |

$45π $ | $2sec45π $ | $-22 ≈-2.83$ |

$47π $ | $2sec47π $ | $22 ≈2.83$ |

$49π $ | $2sec49π $ | $22 ≈2.83$ |

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

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

Explore

Discussion

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

Recall that the $y-$coordinate of this point corresponds to the sine of $θ.$ Therefore, the cosecant function can also be defined as the reciprocal of $sinθ.$

$cscθ=sinθ1 $

Since division by $0$ is not defined, the graph of the parent cosecant function $y=cscx$ has vertical asymptotes where $sinx=0.$ This means that the graph has vertical asymptotes at multiples of $π.$ The graph of $y=cscx$ can be drawn by making a table of values.

Consider the general form of a cosecant function.

$y=acscbx$

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

Properties of $y=acscbx$ | |
---|---|

Amplitude | No amplitude |

Number of Cycles in $[0,2π]$ | $∣b∣$ |

Period | $∣b∣2π $ |

Domain | All real numbers except multiples of $∣b∣π $ |

Range | $(-∞,-∣a∣]∪[∣a∣,∞)$ |

Method

Recall the format of a cosecant function.
*expand_more*
*expand_more*
*expand_more*
*expand_more*

The points found in the table are $(-23π ,-2.83),$ $(-2π ,-2.83),$ $(2π ,2.83),$ and $(23π ,2.83).$
*expand_more*

$y=acscbθ $

Here, $a$ and $b$ are non-zero real numbers and $θ$ is measured in radians. The values of $a$ and $b$ can be used to graph the function. Consider an example function.
$y=2csc21 θ $

In this function, $a=2$ and $b=21 .$ 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

The period of a cosecant function is the quotient of $2π$ and $∣b∣.$

$Period∣b∣2π ⇒∣∣∣ 21 ∣∣∣ 2π =4π $

The period of this function is $4π.$ 2

Graph the Related Sine Function

Consider the related sine function.

$CosecantFunctiony=2csc21 θ Related SineFunctiony=2sin21 θ $

The graph of the sine function can be drawn on a coordinate plane. 3

Graph the Asymptotes and Plot Points

By definition, the asymptotes of the cosecant function occur when the output of the sine function is $0.$ Furthermore, the maximum and minimum points of the sine function are also points on the cosecant function's graph.

4

Divide the Period and Locate Points

Here, the period is $4π$ and the asymptotes occur every $2π$ 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.

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 $-2π$ to $2π$ by making a table of values.

$θ$ | $2csc21 θ$ | $y$ |
---|---|---|

$-23π $ | $2csc(21 (-23π ))$ | $-22 ≈-2.83$ |

$-2π $ | $2csc(21 (-2π ))$ | $-22 ≈-2.83$ |

$2π $ | $2csc(21 ⋅2π )$ | $22 ≈2.83$ |

$23π $ | $2csc(21 ⋅23π )$ | $22 ≈2.83$ |

5

Draw the Graph

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

Example

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.
After doing some calculations, she found the equation of the curve that models the umbrella.
### Answer

### Hint

### Solution

External credits: Kumiko

$y=21 cscx $

Draw the graph of the function for values of $x$ between $-2π$ and $2π.$
Graph the related sine function and recall that the asymptotes of the cosecant function occur at the zeros of the sine function.

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.

$General Formy=acscbxGiven Functiony=21 cscx⇔y=21 csc1x $

Here, $a=21 $ and $b=1.$ The period of the function is the quotient of $2π$ and $∣b∣.$
$Period∣b∣2π ⇒∣1∣2π =2π $

The period of the function is $2π.$ Now consider the related sine function.
$CosecantFunctiony=21 cscx Related SineFunctiony=21 sinx $

This sine function will now be graphed on a coordinate plane for values of $x$ between $-2π$ and $2π.$
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.
Next, more points on the curve of $y=21 cscx$ will be found by making a table of values.

$x$ | $21 cscx$ | $y$ |
---|---|---|

$-35π $ | $21 csc(-35π )$ | $33 ≈0.58$ |

$-34π $ | $21 csc(-34π )$ | $33 ≈0.58$ |

$-32π $ | $21 csc(-32π )$ | $-33 ≈-0.58$ |

$-3π $ | $21 csc(-3π )$ | $-33 ≈-0.58$ |

$3π $ | $21 csc3π $ | $33 ≈0.58$ |

$32π $ | $21 csc32π $ | $33 ≈0.58$ |

$34π $ | $21 csc34π $ | $-33 ≈-0.58$ |

$35π $ | $21 csc35π $ | $-33 ≈-0.58$ |

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

Pop Quiz

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

Closure

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.

These wires are attached to the tower at a height of $5$ meters above the ground. The following function models the lengths of the wires.$y=5cscθ $

Here, $θ$ 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 $4π $ radians with the ground. Round the answer to the nearest integer. {"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formText