Lesson 10

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.

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.

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.)

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.

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.

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).

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.

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.

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).

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.

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 \(2\pi\).