**TANGENT, COTANGENT, COSECANT, AND SECANT FUNCTIONS**

Example 1:

Given , do the following:

a. State the

EXACTperiod.Since

, the period isb = 2. This means that the r/2epresentative pictureisunits in length./2b. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic cotangent function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period./2(3) Divide this period into four equal intervals. Each will be of length

./2 4 = /2 1/4 = /8(4) Draw dashed vertical lines at the beginning and end of the period to represent the

vertical asymptotes(do not draw they-axis as a dashed line).(5) Create the appropriate

representative pictureby using three points marking the intervals.Note that the

x-values are integer multiples ofand the/8y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(/8, 2), (2/8 = /4, 0),.(3/8, 2)(6) Connect the points to form the

representative pictureof the cotangent function.(7) Copy several more cycles in the same manner to the right and left of the r

epresentative pictureon the interval .

- Please note the concavities of the graph of the cotangent function.

- Also, each branch continues to head toward the vertical asymptotes without ever "getting there." Do not draw the branches parallel to the asymptotes.

Note that the units along thex-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!c. Find the

EXACTequations of the twovertical asymptotesthat therepresentative pictureapproaches.The

vertical asymptotesoccur at the beginning and end of therepresentative picture. Knowing theperiodand the location of therepresentative picture, theEXACTequations of the twovertical asymptotesare

andx = 0x = /2

Example 2:

Given is the function , do the following:

a. State the period.

Since

b =, the period is as follows:/4This means that the r

epresentative pictureisunits in length.4b. Sketch the function on the interval

[by hand.8, 8](1) Keep in mind the graph and characteristics of the basic tangent function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period. Note that the r4epresentative pictureof the tangent function is bisected by they-axis!(3) Divide this period into four equal intervals. Each will be of length

.1(4) Draw dashed vertical lines at the beginning and end of the period to represent the

vertical asymptotes(do not draw they-axis as a dashed line).(5) Create the appropriate

representative pictureby using three points marking the intervals.Note that the

x-values are integer multiples ofand the1y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(1, 2), (0, 0),.(1,2)(6) Connect the points to form the

representative pictureof the tangent function.(7) Copy several more cycles in the same manner to the right and left of the

representative pictureon the interval.[8, 8]

- Please note the concavities of the graph of the tangent function.

- Also, each branch continues to head toward the vertical asymptotes without ever "getting there." Do not draw the branches parallel to the asymptotes.

Note that the units along thex-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!c. Find the

EXACTequations of the twovertical asymptotesthat therepresentative pictureapproaches.The

vertical asymptotesoccur at the beginning and end of therepresentative picture. Knowing theperiodand the location of therepresentative picture, theEXACTequations of the twovertical asymptotesare

x =and2x = 2

Example 3:

Given , do the following:

a. State the period.

Since

, the period of our cotangent function is calculated as follows:b = /4

This means that the

representative pictureunits in length.4b. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic cotangent function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period.4(3) Divide this period into four equal intervals. Each will be of length

.1(4) Draw dashed vertical lines at the beginning and end of the period to represent the

vertical asymptotes(do not draw they-axis as a dashed line).(5) Create the appropriate

representative pictureby using three points marking the intervals.Note that the

x-values are integer multiples ofand the1y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

and(1, 1), (2, 0),.(3,1)(6) Connect the points to form the

representative pictureof the tangent function.(7) Copy several more cycles in the same manner to the right and left of the r

epresentative pictureon the interval .

- Please note the concavities of the graph of the cotangent function.

- Also, each branch continues to head toward the vertical asymptotes without ever "getting there." Do not draw the branches parallel to the asymptotes.

Note that the units along thex-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!c. Find the

EXACTequations of the twovertical asymptotesthat therepresentative pictureapproaches.The

vertical asymptotesoccur at the beginning and end of therepresentative picture. Knowing theperiodand the location of therepresentative picture, theEXACTequations of the twovertical asymptotesare

andx = 0x = 4

Example 4:

Given is the function , do the following:

a. State the period.

Since

, the period of our tangent function is calculated as follows:b = 1/2

This means that the representative picture is

units in length.2b. Sketch the function on the interval by hand.

(1) Keep in mind the graph and characteristics of the basic tangent function!

(2) Mark off a distance along the

x-axis starting at the origin to represent the period. Note that the r2epresentative pictureof the tangent function is bisected by they-axis!(3) Divide this period into four equal intervals. Each will be of length

.4 =2/2vertical asymptotes(do not draw they-axis as a dashed line).(5) Create the appropriate

representative pictureby using three points marking the intervals.Note that the

x-values are integer multiples ofand the/2y-values must be calculated using thex-values.Use a calculator!Specifically, the points will be

(/2and, 1), (0, 0),(/2., 1)(6) Connect the points to form the

representative pictureof the tangent function.(7) Copy several more cycles in the same manner to the right and left of the r

epresentative pictureon the interval .

- Please note the concavities of the graph of the tangent function.

x-axis are DIFFERENT from the units along they-axis! As long as you place numbers along your axes it does not matter "how long" your units are!EXACTequations of the twovertical asymptotesthat therepresentative pictureapproaches.vertical asymptotesoccur at the beginning and end of therepresentative picture. Knowing theperiodand the location of therepresentative picture, theEXACTequations of the twovertical asymptotesare

x =andx =

Example 5:

Use Desmos at

https://www.desmos.com/calculatorto draw the graph of .Required Graph Characteristics:

- Show the
representative pictureand copies of one to the left of it and two to the right of it.- Place numbers on both both axes.
- The
x-axis numbers must contain. The beginning/ending point of each interval in each copy of therepresentative picturemust show numbers.- The
y-axis numbers must show the appropriate values for thex-axis numbers used in (3).- Do not make the
y-axis too large. We need to see the concavities clearly!You can find instructions for Desmos at

http://profstewartmath.com/Math127/A_CONTENTS/desmos.htmSince

, the period isb = 2. This means that the r/2epresentative pictureisunits in length. This knowledge will allow us to find the required/2x-values.Considering the value

, thea = 2y-axis can be betweenand6in the Desmos "Graph Settings."6NOTE:

Sometimes, we have to try different settings in the Desmos "Graph Settings" window before the required

x-axis numbers show up. Additionally, when we export the image, we need to determine which size best shows all required characteristics.In the graph above, the Desmos "Graph Settings" for the

x-axis are betweenandand the Step is./8The "Large Square" size was used for Desmos image export. Incidentally, the other settings refused to show all of the

x-axis numbers.