Lesson 10

Beyond $2\pi$

10.1: All the Way Around (5 minutes)

Warm-up

While the idea of rotating beyond one full circle has been hinted at in previous activities, this warm-up is the first time students are asked to think about how to plot a point on the unit circle for a rotation beyond \(2\pi\) radians. Students will continue to make sense of angles beyond \(2\pi\) in the following activities, starting by considering the coordinates of the end of the blade on a windmill.

In the next lesson, students will extend the domain of the cosine and sine functions again to include negative numbers, so it is not necessary to discuss negative angles at this time.

Launch

Tell students to first try marking the points without referencing a unit circle, keeping in mind that one full rotation is \(2\pi\) radians. For additional challenge, ask students to complete the warm-up without referencing a unit circle or using technology.

As an alternative to this activity, ask students to close their books or devices and stand and all face one direction. Tell students to imagine themselves standing at the center of a unit circle facing 0 radians. Display the angle measurements one by one, each time asking students to turn that far. For example, students would make 1 full turn and half of another for \(\frac{6\pi}{2}\) radians. Tell students to return to 0 after each angle.

Student Facing

Here is a unit circle with a point \(A\) marked at \((1,0)\). For each angle of rotation listed here, mark the new location of \(A\) on the unit circle. Be prepared to explain your reasoning.

A circle with center at the origin of an x y plane.
  1. \(B\), \(\frac{\pi}{3}\)
  2. \(C\), \(\frac{4\pi}{3}\)
  3. \(D\), \(\frac{7\pi}{4}\)
  4. \(E\), \(\frac{5\pi}{2}\)
  5. \(F\), \(\frac{6\pi}{2}\)

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display the unit circle with points marked. If students need extra practice working with radians, select students to share strategies they used to place these points, such as by sectioning off the unit circle into equal-sized pieces.

Invite students to share their thinking for the locations of points \(E\) and \(F\). A key point here is that angle measurements can go beyond \(2\pi\) radians. Here are some questions for discussion:

  • “What do you think the cosine value at \(\frac{5\pi}{2}\) radians is? At \(\frac{6\pi}{2}\) radians?” (The value of cosine at these angles is 0 and -1 since these angles correspond to the same point on the unit circle as \(\frac{\pi}{2}\) and \(\pi\) radians, respectively.)
  • “What do you think the sine of \(4\pi\) radians is? \(6\pi\) radians? \(10\pi\) radians?” (The value of sine at all these angles is 0, because they are all different numbers of full rotations and sine has a value of 0 at 0 radians.)

10.2: Going Around and Around and Around (20 minutes)

Activity

The purpose of this activity is for students to make sense of points shown on graphs of cosine and sine as they relate to a simple context: a point at the end of a windmill blade as it rotates counterclockwise. In a previous lesson, students used cosine and sine to determine points on circles. Now that these two functions are “unwrapped,” students connect points on each graph to the position of a point \(P\) on one of the windmill blades.

During the activity, it is important for students to make note of the periodicity of cosine and sine. For example, they identify the angles where point \(P\) is at its highest point in its rotation and notice that this location happens every \(2\pi\) radians. In future lessons, students will consider how to transform cosine and sine to have different periods to match different types of periodic relationships.

Launch

Arrange students in groups of 2. Display an image of a windmill for all to see, such as the one shown here, and leave it up throughout the activity.

Image of windmill with point P on one blade

Tell students that they are going to answer questions about a point \(P\), which is at the end of the blade that starts pointing directly to the right of the center of the windmill (0 radians).

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret the graphs of cosine and sine in the context of a windmill blade. Display only the context of the problem and the two graphs, without revealing the questions that follow. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving values after \(2\pi\) or the location of point \(P\) after a certain number of rotations. This will help students understand the meaning of angle measures larger than \(2\pi\) radians.
Design Principle(s): Maximize meta-awareness; Support sense-making
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content. 
Supports accessibility for: Language; Conceptual processing

Student Facing

The center of a windmill is \((0,0)\) and it has 5 blades, each 1 meter in length. A point \(P\) is at the end of the blade that is pointing directly to the right of the center. Here are graphs showing the horizontal and vertical distances of point \(P\) relative to the center of the windmill as the blades rotate counterclockwise.

Cosine graph. 
Sine graph. 
  1. How many full rotations are shown by the graphs? Explain how you know.
  2. What do the values of the graphs at \(\theta=3\pi\) mean in this context?
  3. List some different angles of rotation that bring \(P\) to the highest point in its circle of rotation. What do you notice about these angles?
  4. How many angles show point \(P\) at a height of 0.71 meters? Explain or show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may struggle thinking about a point going around in a circle, the horizontal location of the point, and the vertical location of the point all at the same time. Encourage these students to start by focusing only on the horizontal location or the vertical location. Students may find it helpful to sketch a unit circle and slowly trace around it (starting from 0 radians) while tracing the wave-like form of one of the trigonometric functions at the same time.

Activity Synthesis

Display the two graphs from the activity for all to see and reference throughout the discussion. Begin the discussion by inviting students to state angles that bring \(P\) to its highest point in its circle of rotation. While students don’t need to generalize to an expression like \(\frac{\pi}{2}+2\pi\boldcdot k\), they should understand that adding any multiple of \(2\pi\) to \(\frac{\pi}{2}\) will give an angle where the blade \(P\) is on is pointing directly up.

Next, select 2–3 students to share their reasoning for the number of angles that show \(P\) at a height of 0.71 meters. If students do not suggest graphing the line \(y=0.71\) to see where it intersects the graph of \(y=\sin(\theta)\), do so and ask students to interpret what the points of intersection mean.

10.3: Back to Where We Started (10 minutes)

Activity

Now that students have considered the meaning of radians greater than \(2\pi\), the purpose of this activity is for them to determine specific values for cosine and sine at these larger angles. In particular, students make connections between angles greater than \(2\pi\) and between 0 and \(2\pi\) that correspond to the same point on the unit circle (MP8).

To complete the table, students may identify the matching angle between 0 and \(2\pi\) and use the approximated values from the unit circle display or they may use technology to calculate the values of cosine and sine for the given angle. Monitor for both approaches to share during the whole-class discussion.

Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. After 2–3 minutes of quiet work time, pause the class and ask students to share their strategies for angles \(\frac {25\pi}{12}\) and \(\frac {5\pi}2\) with their partner before continuing with the rest of the activity.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 3 of the rotations to complete.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

  1. The point \(P\) on the unit circle has coordinates \((1,0)\). For each angle of rotation, state the number of rotations defined by the angle and then identify the coordinates of \(P\) after the given rotation.

    A circle with center at the origin of an x y plane. Point P lies on the outside of the circle, on the x axis, to the right of the origin.
    rotation in radians number of rotations horizontal coordinate vertical coordinate
    \(\frac{3\pi}{2}\) 0.75 0 -1
    \(\frac {25\pi}{12}\)      
    \(\frac {5\pi}2\)      
    \(\frac{7\pi}3\)      
    \(\frac {49\pi}{12}\)      
    \(5\pi\)      
  2. In general, if \(\theta\) is greater than \(2\pi\) radians, explain how you can use the unit circle to make sense of \(\cos(\theta)\) and \(\sin(\theta)\).

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. For each angle of rotation in the activity, determine the minimum number of times you could repeat the rotation to end up back at the point \(P\).
  2. Is it possible to have a rotation that no matter how many times it repeats, you never end up back exactly at \(P\)? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Begin the discussion by selecting students who determined the angle between 0 and \(2\pi\) that matched the given angle in the table and then used the values from a unit circle display to share their reasoning. Then invite students who used technology to determine the coordinates by calculating \(\cos(\theta)\) and \(\sin(\theta)\). Both methods work, but the latter takes advantage of cosine and sine as functions whose domain extends beyond \(2\pi\) radians.

Highlight the values at \(\frac{25\pi}{12}\) and \(\frac{49\pi}{12}\) radians. Both of these angles describe an arc starting and stopping at the same points on the unit circle, but the second completes one extra full rotation before stopping. Remind students that while some values of cosine and sine appear more often than every \(2\pi\) radians, the entire set of values for each repeats every \(2\pi\) radians, and this is why the period of these functions is \(2\pi\).

Representing, Conversing: MLR7 Compare and Connect. As students share their method for filling in the table, either by matching the given angle with the corresponding value in the unit circle or by using technology, invite students to discuss the advantages and disadvantages of each method. For example, ask “Which method do you prefer? Why?”, “Which method is more efficient?”, or “Can both methods be used to find values for any and all angles? Give examples.” Call students’ attention to the different ways of thinking and reasoning presented in the responses. This will help students use critical thinking in discussing the benefits and constraints of determining values by using repeated reasoning on a unit circle or by utilizing technology. 
Design Principle(s): Maximize meta-awareness; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Periodic functions repeat their values at regular intervals. The goal of this discussion is to show how and why the values of cosine repeat. Display this unit circle for all to see:

Unit circle centered at origin 

Tell students that point \(A\) is the result of rotating point \(P\) counterclockwise by \(\frac \pi 2\) radians. It is also the result of rotating point \(P\) counterclockwise by \(\frac {5\pi}2\) radians. Display this graph of \(y=\cos(\theta)\):

Graph of a trigonometric function

Ask students,

  • “How is the point at \((\frac \pi 2, 0)\) related to point \(A\)?” (This point is the horizontal position of \(A\) as it moves around the unit circle, or the cosine of the angle as \(A\) travels around the unit circle. The cosine of \(\frac \pi 2\) is 0, and the horizontal position of point \(A\) is 0 after having been rotated \(\frac \pi 2\)radians from \(P\).)
  • “How is the point at \((\frac {5\pi}2, 0)\) related to point \(A\)?” (Since \(A\) is in the same position on the unit circle after rotating \(\frac \pi 2\) radians and rotating \(\frac {5\pi}2\) radians, its horizontal position is still 0. That is, \(\cos(\frac {5\pi}2)\) must equal \(\cos (\frac \pi 2)\), which is 0.
  • “Which point on the unit circle corresponds to \((\frac{3\pi}{2},0)\)?” (This point is associated with \((0,\text-1)\) on the unit circle, which is where the value of the \(x\)-coordinate goes from negative to positive as \(\theta\) increases. While the \(x\)-coordinate is 0, the \(y\)-coordinate is not the same as when the value of the \(x\)-coordinate goes from positive to negative as \(\theta\) increases.)
  • “How can you tell that \(\cos(\theta)=\cos(\theta+2\pi)\) for any angle \(\theta\)?” (Every time you add \(2\pi\) to the angle, the position of the point is the same on the unit circle. Since the horizontal position of the point is the same, the cosine of the angle is also the same.)

10.4: Cool-down - Turn it Around (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Here is a wheel with radius of 1 foot centered at \((0,0)\). What happens to point \(P\) if we rotate the wheel one full circle? Two full circles? 12 full circles? To someone who didn’t watch us turn the wheel, it would look like we had not moved the wheel at all since all of these rotations take point \(P\) right back to where it started.

A circle, subdivided by 12 congruent central angles. Point P is on the circle on the  right side of the horizontal diagonal. Radius is 1 foot.

Here is a table showing different counterclockwise rotation amounts and the \(x\)-coordinate of the corresponding point.

Notice that when the angle measure increases by \(2\pi\), the \(x\)-coordinate of point \(P\) stays the same. This makes sense because \(2\pi\) radians is a full rotation. In terms of functions, the cosine function, \(\cos(\theta)\), gives the \(x\)-coordinate of the point on the unit circle corresponding to \(\theta\) radians. Since one full circle is \(2\pi\) radians, this means for any input \(\theta\), adding multiples of \(2\pi\) will not change the value of the output. As seen in the table, the value of \(\cos(2\pi)\) is the same as \(\cos(2\pi+2\pi)\), which is the same as \(\cos(2\pi+2\pi+2\pi)\), and so on.

rotations angle measure
in radians
\(x\)-coordinate
of \(P\)
\(\frac{1}{4}\) \(\frac{\pi}{2}\) 0
1 \(2\pi\) 1
\(\frac{5}{4}\) \(\frac{5\pi}{2}\) 0
2 \(4\pi\) 1
\(\frac{9}{4}\) \(\frac{9\pi}{2}\) 0
3 \(6\pi\) 1

We can also see this from the graph of \(y=\cos(\theta)\). To see when \(\cos(\theta)\) takes the value 1, here is a graph of \(y = \cos(x)\) and \(y=1\). Notice that they meet each time the input \(\theta\) changes by \(2\pi\) which makes sense as that represents one full rotation around the unit circle.

A graph.