Degrees or Radians?
This lesson is intended to be taught in degrees. However,
alternative worksheets in radians are also provided. You may use either
version, or even use both versions in the same class as a way to
differentiate for students at various levels of development. Note:
The radian version of this activity should not be used to introduce
radian measure. It is intended for students who already understand the
use of radians.
This lesson is intended for students who have already explored
unit circle trigonometry. In particular, students must understand the
relationship between the coordinates of a point on the unit circle and
the sine and cosine of its central angle in standard position. A quick
review during the pre-assessment stage below is helpful for some
Display the first question on the Pre-Assessment
overhead or draw the diagram on the board. Assign students to label the
lengths of the sides of the right triangle. Create similar examples if
students need extra practice.
Display the second question on the overhead or draw the diagram
on the board. Review and discuss the answers to both questions. For
Question 1, the side lengths are 3, 4, and 5. For Question 2, cos θ should be written on the side on the x-axis and sin θ should be written on the side perpendicular to the x-axis.
Form students into groups of 2 or 3; larger or smaller groups
are not recommended for this activity. This group size will allow
students to collaboratively manipulate the large-scale materials and
promote discussion to solve any problems that may occur. Too many
students in the group may leave some members with little or nothing to
Introduce the activity by telling students that it’s time to
focus on the physical relationships in trigonometric functions.
Students will use a protractor to measure specific angles around the
circle at the beginning of the activity, but no numeric measurements
are used after that. Students will explore the relationships between
lengths by understanding how all the measurements are based only on the
initial spaghetti, which is one unit (hence the unit circle). The focus
for the majority of the lesson is on comparative physical lengths of
spaghetti, not numeric measurements. Some students will need to be
reminded of this throughout the activity.
Distribute the Graphs from the Unit Circle activity sheet to
each student. Depending on the ability level of your students, you can
choose to do this activity in degrees or radians.
Students tend to be more comfortable working in degrees. However, if
your students are fluent in using radians, you can choose to have them
work in those units. Although they are working in groups, stress to
students that each person should turn in a paper. That way each student
will have a copy of the work to keep with his or her notes for future
Do not distribute the Questions activity sheet until students
have finished the Graphs one correctly. The beginning of the Questions
activity sheet essentially answers the questions at the end of the
Graphs activity sheet. It is best if students have a chance to
discover, or remember, these answers.
Instructions for students are on the handout so groups may
begin right away. Various check points are included throughout the
activity. They look like this: OK _________. Students are instructed on
the activity sheet to check with you at each check point to make sure
their work to that point is on track. Initial or stamp the check point
when you feel the student is ready to go on to the next section. This
ensures that each group is interacting with you throughout the activity
and is staying on track. Also, it means you do not have to read those
sections on the students’ papers after you collect them, since you have
already checked them.
The first challenge for students is to figure out how to draw a
circle with a radius equal to one piece of spaghetti. Their usual
compass isn’t big enough. Compasses made for chalkboards or whiteboards
will work, but it is a nice challenge for students to figure out how to
make it themselves. If time is an issue, you may want to give students
a chalk board or whiteboard compass, since it could save 15 or 20
minutes of class time.
As students begin drawing their circles, circulate around the
room to make sure they are well drawn. Be sure to remind students to
mark the center of the circle before drawing it. Also, make sure the
circle is not too sloppy. A lopsided circle will lead to a very
lopsided sine curve and cosine curve.
The initial set-up of the unit circle and function graph on each group’s paper should look like this:
As students begin to measure and mark their angle measures,
watch to be sure they lay the string around the circle in a
counter-clockwise direction starting at (1,0). This reinforces angle
measures in the unit circle for angles in standard position. The graph
will work no matter which direction they put the string on the circle,
but this will help to reinforce what students have already learned
about unit circle trigonometry.
Question 1 on the Graphs activity sheet asks what the x-values
represent on the function graph. At this stage of the process, the
important thing is just for students to realize that they are angle
measures. For students who don’t see it right away, remind them that
they labeled each graph in degrees. Ask students:
- What do we usually measure in degrees?
[Students should recall that angles are measured in degrees.]
- Then, what is labeled on the x-axis?
[Students should conclude that since the x-axis is being labeled in degrees, it most likely represents the angle measures.]
Transferring the spaghetti heights from the unit circle to the function graph looks like this:
Question 2 on the Graphs activity sheet asks what the y-values
represent on the function graph. Students need to recall that this
piece of the triangle in the unit circle is the sine value, which they
reviewed in Question 2 of the Pre-Assessment. Therefore, the y-values
of the function graph are sine values. For a student who doesn’t see it
right away, ask questions related to a previous lesson on unit circle
trigonometry or the Pre-Assessment activity from this lesson. Ask, "How
did you label that leg of the triangle in the pre-assessment activity?"
After initialing or stamping the checkpoint at the end of the
Graphs activity sheet, give each student the Question activity sheet.
Based on which Graphs activity sheet you used, each student should be
given either the degrees or radians version of the activity sheet.
In Questions 7–10, students are challenged to determine what length in the unit circle is used to create y = cos x
on the function graph. For students who struggle to pick the correct
measure ask, "Where did you put the cos θ label in the pre-assessment
activity?" Students should recall that cos θ represented the side of
the triangle along the x-axis.
Students may need help to realize that the horizontal leg of the
triangle from the unit circle has to be placed vertically on the
function graph since cosine is the x-value in a unit circle but the y-value in the graph of y = cos x.
As students complete the activity collect the activity sheets
and allow students to hang their graphs on the wall. You may wish to
find one graph that is particularly neat and accurate to put at the
front of the room as a resource during future lessons.
Post instructions for the closure activity so that groups
finishing the activity at different times can begin this assignment as
Draw a graph of y = sin x in your notebook, this time including 2 complete periods, from x = –360° to x = 360°. Do the same for y = cos x on a different set of axes. Describe the domain, range, maxima, minima, and intercepts of each graph.
This assignment is designed to assess how much students understand these key features of the graphs. Note:If working in radians, the graphs should be drawn from x = –2 radians to x = 2 radians.
On the last day of the lesson, return the handouts to students
for a discussion of the Questions from the Unit Circle activity sheet.
Refer to the Answer Key for sample answers. Randomly select students to answer each question, referring to the graphs on the wall as needed.
- Use the check points to facilitate ongoing assessment. Later,
determine some number of points for each section in order to assign a
grade for the assignment.
- Show students graphs created by other students or graphs
you've created with mistakes and ask them to correct them. For example,
flip the sine graph vertically across the x-axis and ask students how they know it is wrong.
- Assign a journal-writing task in which students explain the connection between the unit circle graph and the function graph.
Similar activities can be done for tangent, cotangent, secant, and
cosecant. You can introduce the other trigonometric functions using the
same activity, or have students create the graphs on their own on a
Questions for Students
1. What happens to the sine value on the unit circle at 180°? How will you show that on the function graph?
[The circle intersects the x-axis, so the sine is 0. Therefore, it should be a zero of the function or an x-intercept of the graph.]
2. What happens to the cosine value on the unit circle at 180°? How will you show that on the function graph?
[Cosine is the x-value in the unit circle, so that would be –1. At 180° on the function graph, the y-coordinate should be –1, so this is a minimum point on the function graph.]
3. What happens to the cosine value on the unit circle at 90°? How will you show that on the function graph?
[The x-value on the unit circle for a 90° angle in
standard position is 0, so the cosine is 0 at 90°. It should be a 0 of
the function or an x-intercept of the graph.]
- Was students’ level of enthusiasm/involvement high or low? Explain why.
- Did you challenge the achievers to go beyond quick answers and pursue deeper explanations and understanding? How?
- Did all members of each group contribute to the project and
remain fully engaged? How do you know? How could you improve that next
- Did you provide appropriate support for students who struggle with graphing by hand? How?
- How did students demonstrate understanding of the relationship between the unit circle and the function graph?
- Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were the effective?