**TANGENT, COTANGENT, COSECANT, AND SECANT FUNCTIONS
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Memorize the shape and characteristics of the graph of the basic tangent function.

2. Memorize the shape and characteristics of the graph of the basic cotangent function.

3. Find the period of transformations of the basic tangent and cotangent functions.

4. Sketch certain transformations of the basic tangent and cotangent functions.

5. Memorize the shape and characteristics of the graph of the basic cosecant function.

6. Memorize the shape and characteristics of the graphs of the basic secant function.

In the previous lesson, we looked at the basic sine and cosie functions and their graphs. We also examined some transformations. In this lesson we will study the basic tangent and cotangent functions, their graphs, and some transformations. Lastly, there will be a brief overview of the basic secant and cosecant functions.

The Basic Tangent Function and its GraphThe tangent function is also called "periodic" because its graph also consist of identical pictures that repeat forever.

The domain contains all real numbers except for all number of the form , where

is any integer. There, the tangen function is undefined.kWe will use the Desmos online graphing calculator to make a picture of .

You can find instructions on how to use Desmos at http://profstewartmath.com/Math127/A_CONTENTS/desmos.htm

YOU MUST MEMORIZE THE SHAPE AND CHARACTERISTICS OF ITS GRAPHGiven , we can create the following table using several

x-values (in radians) from the picture above.

NOTE:

etc. (use your calculator, if necessary!)Wherever the tangent is undefined, the graph of the tangent function will have

vertical asymptotes***. The general equation of thesevertical asymptotesis, wherex =is any integer.k

*** From your algebra course you should know thatvertical asymptotesare invisible vertical lines that a graph cannot touch or cross.We will now plot these points and connect them keeping in mind the shape of the graph. The equations of the

vertical asymptotesare and .We call this portion of the graph of the tangent function its

representative picture. It lies on the intervalwhich is called thePERIODof the tangent graph. Clearly, it is in length.The

representative picturerepeats forever along the positive and negativex-axis and creates the graph of the basic tangent function.The graph is concave up to the right of the

x-intercept and concave down to its left.Note that the

representative pictureis divided intofour equalintervals.

- The graph starts with a
vertical asymptote. The graph never touches thevertical asymptote, but is also never parallel to it.- At the end of the first interval, we find an ordered pair and plot it.
- At the end of the second interval, the graph has an
x-intercept.- At the end of the third interval, we find an ordered pair and plot it.
- At the end of the fourth interval, there is a
vertical asymotote. The graph never touches thevertical asymptote, but is also never parallel to it.Below is a hand-drawn graph of the tangent function. Note that in a hand-drawn graph, we plot the

vertical asymptotesas dashed vertical lines. In a computer-generated graph the asymptotes are not visible.

The Basic Cotangent Function and its GraphThe cotangent function is also called "periodic" because its graph also consist of identical pictures that repeat forever.

The domain contains all real numbers except for all number of the form

, wherekis any integer. There, the cotangent function is undefined.kWe will use the Desmos online graphing calculator to make a picture of .

You can find instructions on how to use Desmos at http://profstewartmath.com/Math127/A_CONTENTS/desmos.htm

YOU MUST MEMORIZE ITS SHAPE AND CHARACTERISTICS!Given , we can create the following table using several

x-values (in radians) from the picture above.

NOTE:

etc. (use your calculator, if necessary!)Wherever the cotangent is undefined, the graph of the cotangent function will have

vertical asymptotes. The general equation of thesevertical asymptotesis, whereis any integer.kWe will now plot these points and connect them keeping in mind the shape of the graph. The equation of the

vertical asymptoteis (they-axis) and .We call this portion of the graph of the cotangent function its

representative picture. Therepresentative picturelies on the intervalwhich is called the(0, )PERIODof the cotangent graph. Clearly, it is in length.The

representative picturerepeats forever along the positive and negativex-axis and creates the graph of the basic cotangent function.Please note the difference between the representative picture of the tangent and that of the cotangent!The graph is concave up to the left of the

x-intercept and concave down to its right.Note that the

representative pictureis divided into four equalintervals.

- The graph starts with a
vertical asymptote. The graph never touches thevertical asymptote, but is also never parallel to it.- At the end of the first interval, we find an ordered pair and plot it.
- At the end of the second interval, the graph has an
x-intercept.- At the end of the third interval, we find an ordered pair and plot it.
- At the end of the fourth interval, there is a
vertical asymotote. The graph never touches thevertical asymptote, but is also never parallel to it.Below is a hand-drawn graph of the cotangent function. Note that in a hand-drawn graph, we plot the

vertical asymptotesas dashed vertical lines. In a computer-generated graph the asymptotes are not visible.

Transformations of the Basic Tangent and Cotangent Functions- see #1 through 5in the "Examples" documentNext, we will investigate

transformationsof the basic tangent and cotangent functions. However, only the functions of the formandy = a tan(bx), wherey = a cot(bx)andaare real numbers withbandaNOT equal to 0.bThe transformations will then be described as follows:

The number

aindicates a vertical stretch or compression of the graph of the basic tangent and cotangent functions. There is no amplitude involved as there is in the sine and cosine functions.

The number

bindicates a horizontal stretch or compression of the graph of the basic tangent and cotangen functions. This will affect the length of the

period.If

is betweenband0, the period of the basic tangent and cotangent function gets larger.1If

is greater thanb, the period of the basic tangent and cotangent function gets smaller.1The formula calculates the period

of the graph of the tangent and cotangent functions.P

Strategy for Sketching Some Tangent and Cotangent Functions- see #1 through 5in the "Examples" documentWe will only sketch functions of the form

andy = a tan(bx).y = a cot(bx)

- Determine the
period.- Keep in mind the graph and characteristics of the basic function.
- Mark off a distance along the
x-axis to represent theperiodof therepresentative picture. Keep in mind that the representative pictureof the tangent function is bisected by they-axis!- Divide the
periodinto four equal intervals.- 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).- Create the appropriate
representative pictureby using three points marking the intervals.- Connect the points in (5) to form the
representative picturebeing mindful of thevertical asymptotes.- Copy the
representative pictureseveral more times along the negative and positivex-axis.

The Basic Cosecant Function and its Graph

The cosecant function is also called "periodic" because its graph also consist of identical pictures that repeat forever.

The domain contains all real numbers except for all number of the form

, wherekis any integer. There, the cosecant function is undefined.kBelow is a hand-drawn graph of the cosecant function together with the sine function. Note that in a hand-drawn graph, we plot the

vertical asymptotesas dashed vertical lines. In a computer-generated graph the asymptotes are not visible.

YOU MUST MEMORIZE ITS SHAPE AND CHARACTERISTICS!Wherever the cosecant is undefined, the graph will have

vertical asymptotes. The general equation of thevertical asymptoteis , whereis any integer.kPlease observe the following:

- The
of the cosecant function lies on the intervalrepresentative picture.(0, 2)- The parabolic branches are separated by
vertical asymptotes, which occur at thex-intercepts of the sine function because that is where the sine value is 0.- The branches never touch the asymptotes, but they are also never parallel to them.
- There are neither
x- nory-intercepts.- The branches have valleys where the graph of the sine function has peaks. Likewise, the branches have peaks where the graph of the sine function has valleys.

The Basic Secant Function and its GraphThe secant function is also called "periodic" because its graph also consist of identical pictures that repeat forever.

The domain contains all real numbers except for all number of the form , where

is any integer. There, the secant function is undefined.kBelow is a hand-drawn graph of the secant function together with the cosine function. Note that in a hand-drawn graph, we plot the

vertical asymptotesas dashed vertical lines. In a computer-generated graph the asymptotes are not visible.YOU MUST MEMORIZE ITS SHAPE AND CHARACTERISTICSWherever the secant is undefined, the graph will have

vertical asymptotes. The general equation of thevertical asymptoteis, wherex =is any integer.kPlease observe the following:

- The
of the secant function lies on the intervalrepresentative picture.[0, 2]- The parabolic branches are separated by
vertical asymptotes, which occur at thex-intercepts of the cosine function because that is where the cosine value is 0.- The branches never touches the asymptotes, but they are also never parallel to them.
- There are neither
x- nory-intercepts.- The branches have valleys where the graph of the cosine function has peaks. Likewise, the branches have peaks where the graph of the cosine function has valleys.

A Word about Transformations of the Basic Cosecant and Secant FunctionsJust like for the sine, cosine, tangent, and cotangent functions, transformations are also possible for the basic cosecant and secant functions.

The formula calculates the period

of the graph of the cosecant and secant functions.P