Secant and Cosecant
Graphing Activity
This activity will get your students to understand difficult concepts about secant and cosecant graphs:
 Where does this graph even come from?
 Why does the graph repeat itself infinitely?
 How do different numbers effect the graph when they are inserted into different places in the equation?
This is one of my favorite activities to do in the trigonometry unit. After students learn about the unit circle,
including positive and negative angles in radian form and graphing sine & cosine waves, this activity creates the
perfect transition to graphing secant and cosecant functions. The activity is laid-out step-by-step so that
students can follow the instructions without a teacher by their side. It is an outstanding small group activity, but
I have used it as a homework assignment, as well.
BEFORE CLASS:
• Make copies of the pages.
• Assign students to groups (if you are using this as a small group activity)
DURING CLASS:
• Pass out worksheets to students
• Walk around the class & eavesdrop on the educational conversations that take place
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Secant and Cosecant Graphing Activity
Your wonderful teacher is giving you the amazing opportunity to review important facts about trig functions while also learning something new! In the
chart below, you should do each of the following:



Write the smallest positive coterminal angle associated with each negative angle
Write the sine of each angle in decimal form. (Round to the nearest .01 units for this assignment.)
Remembering the inverse relationship between sine and cosecant, write the cosecant of each angle in decimal form.
Coterminal
angle
X
X
X
X
X
X
X
X
angle
1. What strange thing happened to the cosecant at
(along with some other places, perhaps)?
2. When making a graph, what does it look like when you plot this?
3. Graph each of the points
each of the points
on the COSECANT graph that was given to you, connecting them with a DOTTED line. Then graph
on the SAME graph, connecting them with a SOLID line.
4. What is the period of the cosecant graph?
5. What similarities do you notice between the sine and cosecant graphs?
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X
Now, fill in the boxes below using cosine and secant.
Coterminal
angle
X
X
X
X
X
X
X
X
X
angle
6. Graph each of the points
on the SECANT graph that was given to you, connecting them with a DOTTED line. Then graph each
of the points
on the SAME graph, connecting them with a SOLID line.
7. What is the period of the secant graph?
8. List three similarities between the secant and cosecant graphs.
9. List three differences between the secant and cosecant graphs.
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Secant and Cosecant Graphing Calculator Activity
On your graphing calculator, press the WINDOW button. Enter the following values. XMin: -2 ; XMax: 2 ; XScl: ; YMin: -5; YMax: 5; YScl: 1.
Then press Y= and type the equation: 1/ sin(1x - 0) + 0. Next, press GRAPH. Check to make sure that this graph matches the cosecant graph that
you created earlier. y = A / sin(Bx - C) + D is the general form for a cosecant graph. You will now experiment with your graphing calculator to
determine how each of the values change the graph. You may need to graph more than one value to determine the answer to each problem.
1. Change the value of A to any number between 0 and 1. Describe what happens to a cosecant graph when 0 < A < 1.
2. Change the value of A to any negative number. Describe what happens to a cosecant graph when A < 0.
3. Change the value of A to any number larger than 1. Describe what happens to a cosecant graph when A > 1.
4. Change the value of B to any number between 0 and 1. Describe what happens to a cosecant graph when 0 < B < 1.
5. Change the value of B to any negative number. Describe what happens to a cosecant graph when B < 0.
6. Change the value of B to any number larger than 1. Describe what happens to a cosecant graph when B > 1.
7. Change the value of C to any positive number. I recommend a multiple of . (It is important to note that a positive value for C means that your
equation will be in the form of csc(X - #). ) Describe what happens to a cosecant graph when C > 0.
8. Change the value of C to any negative number. (It is important to note that a positive value for C means that your equation will be in the form of
csc(X + #). ) Describe what happens to a cosecant graph when C < 0.
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9. Change the value of D to any positive number. Describe what happens to a cosecant graph when D > 0.
10. Change the value of D to any negative number. Describe what happens to a cosecant graph when D < 0.
All of the rules that you have written are true for both cosecant and secant. Now, you just need to remember the similarities and differences between
their two graphs. Use the graphs of csc(x) and sec(x) that you drew earlier to answer each of the problems.
11. Which part of csc(x) appears on the y-axis: the local maximum, local minimum, or the asymptote?
12. Which part of sec(x) appears on the y-axis: the local maximum, local minimum, or the asymptote?
13. Is the first full shape after the y-axis smiley or frowny for csc(x)?
14. Is the first full shape after the y-axis smiley or frowny for sec(x)??
15. List some general rules/formulas in each box.
Cosecant
Local Maximums
Local Minimums
Asymptotes
Period
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Secant
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Secant and Cosecant Graphing Activity KEY
Your wonderful teacher is giving you the amazing opportunity to review important facts about trig functions while also learning something new! In the
chart below, you should do each of the following:



Write the smallest positive coterminal angle associated with each negative angle
Write the sine of each angle in decimal form. (Round to the nearest .01 units for this assignment.)
Remembering the inverse relationship between sine and cosecant, write the cosecant of each angle in decimal form.
Coterminal
angle
0
X
X
X
X
X
X
X
X
X
0
0
0
angle
0
.71
1
.71
0
-.71
-1
-.71
.71
1
.71
-.71
-1
-.71
Und. 1.41
1
1.41 Und. -1.41
-1
-1.41 Und. 1.41
1
1.41 Und. -1.41
-1
-1.41 Und.
1. What strange thing happened to the cosecant at
(along with some other places, perhaps)? It was undefined.
2. When making a graph, what does it look like when you plot this?A vertical asymptote gets drawn there
3. Graph each of the points
on the COSECANT graph that was given to you.
4. What is the period of this graph?
5. What similarities do you notice between the sine and cosecant graphs? same period; asymptotes of csc(x) match x-intercepts of sin(x)
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Now, fill in the boxes below using cosine and secant.
Coterminal
angle
0
X X
X
X
X
X
X
X
X
0
angle
1
.71
0
-.71
-1
-.71
0
.71
1
.71
0
-.71
-1 -.71
.71
1
1
1.41
Und.
-1.41
-1
-1.41
Und.
1.41
1
1.41
Und.
-1.41
-1 -1.41 Und. 1.41
1
6. Graph each of the points
7. What is the period of this graph?
on the SECANT graph that was given to you.
8. List three similarities between the secant and cosecant graphs. Answers vary. Some possibilities are:
 vertical asymptotes (spaced pi apart)
 Local maximums all at -1
 Local minimums all at 1
 Smiley/Frowny pattern separated by vertical asymptotes
 Same period
9. List three differences between the secant and cosecant graphs. Answers vary. Some possibilities are:
 vertical asymptotes appear in different places
 Local maximums appear in different places
 Local minimums appear in different places
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Secant and Cosecant Graphing Calculator Activity
On your graphing calculator, press the WINDOW button. Enter the following values. XMin: -2 ; XMax: 2 ; XScl: ; YMin: -5; YMax: 5; YScl: 1.
Then press Y= and type the equation: 1/ sin(1x - 0) + 0. Next, press GRAPH. Check to make sure that this graph matches the cosecant graph that
you created earlier. y = A / sin(Bx - C) + D is the general form for a cosecant graph. You will now experiment with your graphing calculator to
determine how each of the values change the graph. You may need to graph more than one value to determine the answer to each problem.
1. Change the value of A to any number between 0 and 1. Describe what happens to a cosecant graph when 0 < A < 1.
Local maximums decrease and local minimums increase, creating a smaller vertical "gap".
2. Change the value of A to any negative number. Describe what happens to a cosecant graph when A < 0.
The "smileys" and "frownys" change places. (X-axis reflection)
3. Change the value of A to any number larger than 1. Describe what happens to a cosecant graph when A > 1.
Local maximums increase and local minimums decrease, creating a wider vertical "gap".
4. Change the value of B to any number between 0 and 1. Describe what happens to a cosecant graph when 0 < B < 1.
The "smileys" and "frownys" become wider, and vertical asymptotes become farther apart.
5. Change the value of B to any negative number. Describe what happens to a cosecant graph when B < 0.
The "smileys" and "frownys" change places. (Y-axis reflection)
6. Change the value of B to any number larger than 1. Describe what happens to a cosecant graph when B > 1.
The "smileys" and "frownys" become thinner, and vertical asymptotes become closer together.
7. Change the value of C to any positive number. I recommend a multiple of . (It is important to note that a positive value for C means that your
equation will be in the form of csc(X - #). ) Describe what happens to a cosecant graph when C > 0.
The graph shifts to the right by the amount that gets entered for C.
8. Change the value of C to any negative number. (It is important to note that a positive value for C means that your equation will be in the form of
csc(X + #). ) Describe what happens to a cosecant graph when C < 0.
The graph shifts to the left by the amount that gets entered for C.
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9. Change the value of D to any positive number. Describe what happens to a cosecant graph when D > 0.
The graph shifts up by the amount that gets entered for D.
10. Change the value of D to any negative number. Describe what happens to a cosecant graph when D < 0.
The graph shifts down by the amount that gets entered for D.
All of the rules that you have written are true for both cosecant and secant. Now, you just need to remember the similarities and differences between
their two graphs. Use the graphs of csc(x) and sec(x) that you drew earlier to answer each of the problems.
11. Which part of csc(x) appears on the y-axis: the local maximum, local minimum, or the asymptote?
The asymptote
12. Which part of sec(x) appears on the y-axis: the local maximum, local minimum, or the asymptote?
The local minimum
13. Is the first full shape after the y-axis smiley or frowny for csc(x)? Smiley
14. Is the first full shape after the y-axis smiley or frowny for sec(x)? Frowny
15. List some general rules/formulas in each box.
Cosecant
Secant
Local Maximums
|A|+D
|A|+D
Local Minimums
-|A|+D
-|A|+D
Asymptotes
Period
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y = csc(x)
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y = sec(x)
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Thank you very much for your purchase! I hope you enjoy this activity! Please give me
feedback & a rating! If you have any requests, comments, or corrections, please send
me a message. At Teachers Pay Teachers, search for DAVID ROBERTSON or use this
link:
http://www.teacherspayteachers.com/Store/David-Robertson
E-mail me at [email protected]
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