Lesson 11

Extending the Domain of Trigonometric Functions

11.1: Rewind to the Windmill (5 minutes)

Warm-up

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.

Launch

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:

Image of windmill with point P on one blade

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. 

Student Facing

Priya is thinking about the windmill in an earlier lesson. That windmill had a point \(P\) at \((1,0)\) at the end of the blade that starts at 0 radians pointing directly to the right. Priya says that if the blade rotates \(\text- \frac \pi 2\) radians, then \(P\) will be at the lowest point in its circle of rotation.

What do you think Priya means by rotating \(\text- \frac \pi 2\) radians? Do you agree with Priya? Be prepared to explain your reasoning.

Student Response

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

Activity Synthesis

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.)

11.2: Math Talk: The Hour Hand (10 minutes)

Activity

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.

Launch

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)
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Here is a clock showing an hour hand at 3 p.m.

A clock on a grid. The hour hand is pointing to the 3. 

Your teacher will give you a time. Identify the radian angle measure that the hour hand rotates through if it starts at 3 p.m.

 

Student Response

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

Activity Synthesis

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.

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

11.3: The Big Picture for Cosine and Sine (15 minutes)

Activity

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).

Launch

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

Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. During the launch take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: maximum value of the function, minimum value of the function, and period.
Supports accessibility for: Conceptual processing; Language

Student Facing

  1. Create a visual display for the following functions. Include a graph of the function from at least \(\text-4\pi\) to \(4\pi\) radians, the maximum and minimum value of the function, and the period of the function. Label any intersections the graph of the function has with the axes.
    1. \(y=\cos(\theta)\)
    2. \(y=\sin(\theta)\)
  2. The \(y\)-axis is a line of symmetry for one of the two graphs. Which one? Explain how you know.

Student Response

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

Student Facing

Are you ready for more?

Recall that a symmetry is a rigid transformation that takes a figure onto itself (not counting a transformation that leaves every point where it is). For example, the graph of \(y=(x-3)^2\) has reflection symmetry across the line \(x=3\).

  1. Find a reflection symmetry of the graph of \(y=\sin(\theta)\).
  2. Find a rotation symmetry of the graph of \(y=\sin(\theta)\).
  3. Find a translation symmetry of the graph of \(y=\sin(\theta)\).

Student Response

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

Anticipated Misconceptions

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.

Activity Synthesis

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.

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. Invite students to quietly circulate and read at least 2 other visual displays in the room. Give students quiet think time to consider what is the same and what is different about the questions and displays. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that include mathematical language and reasoning about cosine and sine functions.
Design Principle(s): Cultivate conversation

11.4: Cosine and Sine Together (15 minutes)

Optional activity

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).

Launch

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing technology. Some students may benefit from a checklist or list of steps to be able to use the technology. 
Supports accessibility for: Organization; Conceptual processing; Attention

Student Facing

Use graphing technology to graph the functions \(y = \cos(\theta)\) and \(y = \sin(\theta)\) on the same axes.

  1. Identify two points where the graphs intersect—one with a negative \(\theta\)-coordinate, and one with a positive \(\theta\)-coordinate. What is the exact \(\theta\)-coordinate for each point? Explain or show how you know.
  2. What are the \(y\)-coordinates of the points of intersection? Explain or show how you know.
  3. What could be the value of \(\cos(\theta)\) if \(\sin(\theta)=0\)? Explain your reasoning.

Student Response

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

Anticipated Misconceptions

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 \(\theta\).

Activity Synthesis

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

Graph of 2 trigonometric functions 

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.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their verbal and written responses to the question, “What are the \(y\)-coordinates of the points of intersection? Explain or show how you know?” Give students time to meet with 2–3 partners to share and get feedback on their responses. The listener should press for detail and precision in language by asking “How does your method relate to the unit circle?”, “Can you make any connections to the Pythagorean Identity?”, or “How can you find another point of intersection that is not on the graph?” Invite students to go back and revise or refine their written explanation based on the feedback from peers. This will help students explain their method for finding coordinates of the points of intersection for the graphs of cosine and sine.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is to directly address a possible point of confusion for students as they transition between thinking about the coordinates of points on a unit circle and the input-output pairs on the graphs of cosine and sine. In particular, this discussion addresses how the naming of the outputs can lead to confusion due to the common use of \(y\) as the output variable.

Display the graphs of \(y=\cos(\theta)\) and \(y=\sin(\theta)\) for all to see. Here are some questions for discussion:

  • “What do the \(y\)-values in the graph of \(y=\cos(\theta)\) represent?” (The \(x\)-coordinate of a point on the unit circle rotated \(\theta\) radians counterclockwise from \((1,0)\).)
  • “What do the \(y\)-values in the graph of \(y=\sin(\theta)\) represent?” (The \(y\)-coordinate of a point on the unit circle rotated \(\theta\) radians counterclockwise from \((1,0)\).)

A key point to highlight here is that it is easy to use the same letters to mean different things, particularly \(x\) and \(y\). To avoid mix-ups, an important part of reading any graph is understanding what is being graphed. (It’s also okay to relabel axes as you work to make things clearer.) In the case of functions, we need to understand the input and the output and what they represent. So far, we have only considered cosine and sine functions that have inputs of radian angles. In future lessons, we will consider transformations of these functions that can accept a variety of input types in order to model different periodic relationships.

11.5: Cool-down - Turn it Around Again (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

The functions \(\cos(\theta)\) and \(\sin(\theta)\) are both periodic, meaning their values repeat at regular intervals. Since the period of cosine and sine is \(2\pi\), the values of these functions repeat any time the input is changed by a multiple of \(2\pi\). We can see this in the graph of \(y=\sin(\theta)\) shown here.

Notice that between \(\text-2\pi\) and \(0\), and then between 0 and \(2\pi\), the same wave pattern repeats. Both positive and negative values for \(\theta\) can be thought of through the lens of the unit circle with positive values indicating counterclockwise rotation and negative values indicating clockwise rotation. This means that different important features of the graph occur at regular intervals:

  • The \(\theta\)-intercepts of the graph are at all integer multiples of \(\pi\).
  • The relative maximums are at \(\frac{\pi}{2}\) and any integer multiple of \(2\pi\) from there.
  • The relative minimums are at \(\frac{3\pi}{2}\) and any integer multiple of \(2\pi\) from there.
Graph. 

The graph of \(y= \cos(\theta)\) is also periodic, repeating every time the input changes by a multiple of \(2\pi\).

  • The \(\theta\)-intercepts are at \(\frac{\pi}{2}\) and any integer multiple of \(\pi\) from there.
  • The relative maximums are at 0 and any integer multiple of \(2\pi\) from there.
  • The relative minimums are at \(\pi\) and any integer multiple of \(2\pi\) from there.
Graph.