Rich – AAT(H)
Name: _________________________________
Spaghetti Lab: Trig Graphs
Date: ___________________ Period: _________
Essential Question: Can I identify sine and cosine graphs based on their shape?
Learning Targets: Students will be able to…
 3.B.1: sketch the graphs of basic sine and cosine functions
Materials: Each day, place your materials in your bag with group member’s names on it.
 7 pieces of spaghetti, one broken and
 Compass
colored by the teacher
 Markers or colored pencils
 Piece of string
 Ziploc bag
 Piece of butcher paper
 Permanent marker
 Ruler
 Protractor
Setup:
1. Using the colored piece of spaghetti to represent the length of the radius of a circle, draw a circle on
the far left side of the butcher paper. This colored spaghetti’s length represents 1 “spaghetti unit”.
2. Draw 2 coordinate plane axes, one above the other, to the right of the unit circle. For each of these
axes, the y-axis should be at least 1 “spaghetti unit” long both above and below the x-axis, and the
x-axis should be at least 6.5 “spaghetti units” long. Your butcher paper should look something like
this:
3. Using a protractor to determine angle measures, mark every 15 degrees around the circle. Label
each mark with its positive angle measure. Draw the x-axis by connecting the 0° and 180° marks.
4. Place the string along the circle with one end at 0°. Transfer all of the 15° marks from the circle to
the string.
5. Stretch the string on each of the x-axes and transfer the marks on the string to the axes. The end of
the string that was at 0° must be placed at the origin. Label each mark with its positive angle
measure.
6. Label each of the x-axes as “Angle in Degrees”. Label the y-axis of the top graph as “ y  cos  x  ”
and the y-axis of the bottom graph as “ y  sin x  ”.
CHECKPOINT! Have Mrs. Rich initial that your set-up is correct.
_____________________
Activity: You are now going to create the graphs of y  sin x  and y  cos  x  using the unit circle and right
triangles.
Procedure:
1. Use the spaghetti to form a standard right triangle on the circle. The hypotenuse of the triangle will
be the radius of the circle from the origin to the 15° mark. Break the spaghetti into the appropriate
lengths for the legs of the triangle.
2. Since this is a unit circle, the length of the horizontal leg of the triangle is equal to cos 15 . Move
the piece of spaghetti that was the horizontal leg of the triangle to the top axes where you are
graphing cosine.
3. Place the piece of spaghetti perpendicular to the x-axis at 15°, with one end of it on the x-axis and
the other above the axis. Make a dot on the paper at the top of the piece of spaghetti to show the
length of the horizontal leg of the 15° triangle.
4. Since this is a unit circle, the length of the vertical leg of the triangle is equal to sin15 . Move the
piece of spaghetti that was the vertical leg of the triangle to the bottom axes where you are
graphing sine.
5. Place the piece of spaghetti perpendicular to the x-axis at 15°, with one end of it on the x-axis and
the other above the axis. Make a dot on the paper at the top of the piece of spaghetti to show the
length of the vertical leg of the 15° triangle.
6. Repeat the process described in steps 1 – 5 for all of the angles in the first quadrant (there should
be a total of 5).
CHECKPOINT! Have Mrs. Rich initial that your work is correct.
_____________________
7. Repeat the process described in stems 1 – 5 for the rest of the angles on the unit circle. Some things
to keep in mind…
 Remember that the triangles collapse at the quadrantal angles (0°, 90°, 180°, 270°, 360°,
etc.), so think about how we deal with that.
 Remember that sine and cosine can become negative as we move past the first quadrant, so
think about how you should plot those points. For example, in the second quadrant, cosine
becomes negative, so that needs to be represented in its graph.
 Try to use as few spaghetti lengths as possible – they can be reused often!
CHECKPOINT! Have Mrs. Rich initial that your graphs are correct.
_____________________
8. Connect the dots on each of the graphs to create a smooth curve.
9. Complete the lab write-up as a group. This will be graded and combined with 6 points of
participation based on the rubric below.
0 points
- All students did not work
effectively and collaboratively
 Off task behavior
- Not all check points have
been met according to Mrs.
Rich
- Materials were not used
appropriately
- Graphs are not complete
and poorly done
- Visual representation does
not meet expectations
2 points
- Most students did not work
effectively and collaboratively
 Some off task behavior
- Not all check points have
been met according to Mrs.
Rich
- Materials were not used
appropriately
- Graphs are not complete
- Visual representation is
simple
4 points
- Most students worked
effectively and collaboratively
 Mostly on task behavior
- All check points have been
met according to Mrs. Rich
- Materials were used
appropriately
- Graphs are complete
- Nice visual representation;
meets expectations
6 points
- All students worked
effectively and collaboratively
 On task behavior
- All check points have been
met according to Mrs. Rich
- Materials were used
appropriately
- Graphs are neat and
complete
- Visual representation of
graph exceeded expectations
Names: _________________________________________________________________________________
Spaghetti Lab: Trig Graphs Follow-up Questions (44 points)
Total: ______/50
Answer the following questions to clarify patterns seen and concepts learned during the lab. Answer all
questions in complete sentences. Each group only needs to turn in one write-up.
1. (13 points) Use your sine graph to answer the following questions.
a. At 0°, what was the y-coordinate? Why did the graph begin at this coordinate point?
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b. Where did the graph go from there? Explain the progression of the graph from 0° to 360°.
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c. What was the maximum value of sine and at what degree marks did it occur?
Max: ______________ at _________________
Why? ____________________________________________________________________________
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d. What was the minimum value of sine and at what degree marks did it occur?
Min: ______________ at _________________
Why? ____________________________________________________________________________
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e. What were the x-intercepts of the graph?
____________________________________________________________________________
f. When was the graph positive? When was it negative? Why?
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g. Describe the graph’s symmetry.
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2. (13 points) Use your cosine graph to answer the following questions.
a. At 0°, what was the y-coordinate? Why did the graph begin at this coordinate point?
_________________________________________________________________________________
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b. Where did the graph go from there? Explain the progression of the graph from 0° to 360°.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
c. What was the maximum value of cosine and at what degree marks did it occur?
Max: ______________ at _________________
Why? ____________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
d. What was the minimum value of cosine and at what degree marks did it occur?
Min: ______________ at _________________
Why? ____________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
e. What were the x-intercepts of the graph?
____________________________________________________________________________
f. When was the graph positive? When was it negative? Why?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
g. Describe the graph’s symmetry.
_________________________________________________________________________________
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3. (3 points) What is the definition of the period of a graph?
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a. After how many degrees do the graphs of sine and cosine start to repeat? __________________
b. What is the period of the sine curve? __________________
c. What is the period of the cosine curve? __________________
4. (2 points) How would you continue the graph of sine or cosine past 360°? Explain the process.
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5. (3 points) Compared with the radius, what is the height of the triangle at…
a. 30°? _________ b. 150°? _________ c. 330°? ____________ d. 570°? ____________
6. (2 points) If you build triangles only at the 15°, 30°, and 45° marks around the unit circle, what is the
smallest number of different triangles that you need to form in order to obtain the lengths needs to
graph one period of the sine or cosine curve? Justify your answer.
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7. (2 points) Explain the relationship between sin 30 and sin150 . How does these differ from
sin210 and sin 330 ? _______________________________________________________________
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8. (2 points) Explain the relationship between cos  30 and cos 150 . How does these differ from
cos 210 and cos  330 ? ______________________________________________________________
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9. (2 points) Explain the relationship between cos  45 and sin 45  . Why does this occur? How does
this differ in each quadrant? _____________________________________________________________
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10. (2 points) Explain the relationship between sin 30 and cos  60 . Why does this occur? How does
this differ in each quadrant? _____________________________________________________________
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