Lesson 11

In this warm-up, students consider the meaning of negative angles on the unit circle and how to interpret the cosine or sine of a negative angle. In the following activity, students will use their thinking here to identify specific negative radian angles associated with rotations of the hour hand of a clock.

If needed, display this image of the windmill from an earlier activity, reminding students that the radius of the circle \(P\) rotates around is 1 meter:

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner. Follow with a whole-class discussion. 

Select students to share their thinking about what Priya means. The important point here is that positive radian measure is a counterclockwise rotation on the unit circle, so we can think of negative radian measure as a clockwise rotation on the unit circle.

Here are some questions for discussion:

• “If point \(P\) is 1 meter from the center of rotation, what does the value of \(\cos(\text-\frac{\pi}{2})\) tell us? \(\sin(\text-\frac{\pi}{2})\)?” (Since \(P\) is at the lowest point at \(\text-\frac{\pi}{2}\), the value of \(\cos(\text-\frac{\pi}{2})\) tells us the \(x\)-coordinate at this position lines up with the \(x\)-coordinate of the center and the value of \(\sin(\text-\frac{\pi}{2})\) tells us the \(y\)-coordinate is 1 meter below the center.)
• “What is another negative angle that would have these same values for the \((x,y)\) coordinates of \(P\)?” (\(\text-\frac{\pi}{2}-2\pi\) radians, since it is one complete clockwise rotation from \(\text-\frac{\pi}{2}\) radians.)

This Math Talk encourages students to think about negative radians and to rely on the structure of a clock face to mentally solve problems. The strategies elicited here build directly from students’ work with positive radians and will be helpful later in the lesson when students study graphs of cosine and sine for both positive and negative angle inputs.

To determine the angle of rotation of the hour hand, students need to look for and make use of structure (MP7). In describing their strategies, students need to be precise in their word choice and use of language (MP6). Depending on the level of challenge appropriate, students may or may not have access to a display of the unit circle throughout this activity. Another option is to allow a unit circle for the first few problems, but then ask students to complete the latter problems without it.

Arrange students in groups of 2. Display the clock image for all to see. Tell students that you are going to name different times and their job is to identify the radian angle measure that the hour hand rotates through if it starts at 3 p.m.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

• 6 p.m.
• 7 p.m.
• 11 p.m.
• 4 a.m. (the next morning)

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

If time allows, continue this activity by asking students to think in the opposite direction. For example, display a negative radian angle and ask students to identify what time it is.

The goal of this activity is for students to create visual displays for the cosine and sine functions that show their graphs for a range of negative and positive radian inputs. As they create their displays, students will also consider what it means for a periodic function to have a maximum or minimum and why \(y=\cos(\theta)\) is symmetric about the \(y\)-axis while \(y=\sin(\theta)\) is not (MP7).

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

Arrange students in groups of 2–4. Provide supplies for making visual displays.

The graphs of cosine and sine have smooth repeating curves, which can be a challenge to draw for students who have not done so before. Encourage students to plot points to help guide their curves as they make the graphs.

Here are some questions for discussion, focusing on some important features of the graph of cosine:

• “What is the period of \(y = \cos(\theta)\)? How do you see this in the graph?” (\(2\pi\), the graph looks like a wave which repeats each time the input is increased or decreased by \(2\pi\).)
• “For which values of \(\theta\) does \(y = \cos(\theta)\) take its maximum value?” (\(0, \pm 2\pi, \pm 4\pi, ...\))
• “How is this related to the period of the cosine function?” (The values of \(y=\cos(\theta)\) repeat each time the input is changed by a multiple of \(2\pi\) so the maximum values are spaced \(2\pi\) apart.)
• “Why does \(y = \cos(\theta)\) have a value of 0 every \(\pi\) radians instead of every \(2\pi\) radians?” (The zeros are equally spaced every \(\pi\) radians because the cosine takes the value 0 twice on the unit circle at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).)

In future lessons, students will learn how to transform these functions to model situations, bringing together their work in a previous unit with their study of trigonometry.

This activity is optional. Use this activity to give students additional practice working with the graphs of cosine and sine.

The goal of this activity is for students to consider the graphs of \(y=\cos(\theta)\) and \(y=\sin(\theta)\) together and reason about both their points of intersection and the value of cosine when \(\sin(\theta)=0\). Students should be encouraged to use their displays of the graphs for these functions and the unit circle and transition between these ways of thinking about trigonometric functions as needed to make sense of the problems (MP7).

If students are struggling identifying an angle \(\theta\) for which \(\cos(\theta) = \sin(\theta)\), ask them to think about the unit circle and what this means about the point on the unit circle representing the angle \(\theta\).

Display the graphs of cosine and sine, such as the one shown here, for all to see and reference throughout the discussion:

Invite students to share how they identified points of intersection between the two graphs, highlighting different strategies for identifying the coordinates, such as using the approximations from the unit circle display or using the Pythagorean Identity to calculate the actual \(y\)-coordinates of the points of intersection. If students did not consider using the Pythagorean Identity to reason about the exact location of the intersection of the two graphs, invite them to do so now.