Lesson 9

Introduction to Trigonometric Functions

9.1: An Angle and a Circle (5 minutes)

Warm-up

The purpose of this warm-up is to prepare students for “unwrapping” the unit circle in the next activity and seeing what the graphs of \(y=\cos(x)\) and \(y=\sin(x)\) look like for the first time in this unit. Part of thinking of cosine and sine as functions is understanding their input and output. The prompts in this activity ask students to informally describe the output of the two trigonometric functions by focusing on each coordinate of a point on the unit circle.

Launch

Display the image for all to see.

Student Facing

Suppose there is a point \(P\) on the unit circle at \((1,0)\).

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.
  1. Describe how the \(x\)-coordinate of \(P\) changes as it rotates once counterclockwise around the circle.
  2. Describe how the \(y\)-coordinate of \(P\) changes as it rotates once counterclockwise around the circle.

Student Response

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

Activity Synthesis

Invite students to share their descriptions, highlighting responses that use actual values such as 1, 0, and -1.

9.2: Do the Wave (15 minutes)

Activity

The goal of this activity is for students to plot the relationship between the values of the \(x\)- and \(y\)-coordinates of points on the unit circle and their corresponding angles from 0 to \(2\pi\). Sometimes known as “unwrapping” the unit circle, students first focus on the \(x\)-coordinate and generate a graph of cosine. They then do the same for the \(y\)-coordinate and generate a graph of sine.

In order to plot the points, students should either reference previously made displays of the unit circle in which the approximate coordinates for the points are listed or use technology to approximate the values of cosine and sine.

Monitor for any students who make the connection between the general shape of the graphs and the descriptions from the warm-up to share during the whole-class discussion.

Launch

Consider demonstrating how to use the applet.  

Note that the increments of the angle measurements are in multiples of \(\frac \pi{12} \), so for example, you may need to point out that \(\frac{5\pi}6\) and \(\frac {10\pi}{12}\) are equivalent.

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 the class display, the table from a previous lesson, or the applet to estimate the value of \(y=\cos(\theta)\) and \(y=\sin(\theta)\) where \(\theta\) is the measure of an angle in radians.

  1. Use technology to plot the values of \(y=\cos(\theta)\), where \(\theta\) is the measure of an angle in radians.

  2. Use technology to plot the values of \(y=\sin(\theta)\), where \(\theta\) is the measure of an angle in radians.

  3. What do you notice about the two graphs?

  4. Explain why any angle measure between 0 and \(2\pi\) gives a point on each graph.

  5. Could these graphs represent functions? Explain your reasoning.

Student Response

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

Launch

Focus student attention on the coordinate axes and make sure they realize that the tick marks are increments of \(\frac \pi{12}\) radians—the same increments for the angles they worked with in the previous lesson.

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

  1. For each tick mark on the horizontal axis, plot the value of \(y=\cos(\theta)\), where \(\theta\) is the measure of an angle in radians. Use the class display of the unit circle, the unit circle from an earlier lesson, or technology to estimate the value of \(\cos(\theta)\).

    A blank graph. 
  2. For each tick mark on the horizontal axis, plot the value of \(y=\sin(\theta)\). Use the class display of the unit circle, the unit circle from an earlier lesson, or technology to estimate the value of \(\sin(\theta)\).

    A blank graph. 
  3. What do you notice about the two graphs?

  4. Explain why any angle measure between 0 and \(2\pi\) gives a point on each graph.

  5. Could these graphs represent functions? Explain your reasoning.

Student Response

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

Anticipated Misconceptions

If students are unsure where to start graphing, recommend that they start by looking at the value of cosine at the angles 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\).

Activity Synthesis

Display completed graphs for cosine and sine from the activity for all to see alongside a unit circle for reference throughout the discussion. Begin the discussion by selecting previously identified students to share their observations. If no one made the connection to the warm-up descriptions and the current graphs, invite students to do so now, highlighting how the graph of cosine matches the description for the \(x\)-coordinate and the graph of sine matches the description for the \(y\)-coordinate.

Next, invite students to share whether they think these graphs represent functions. To help convince students they are, ask, “What are the inputs? The outputs?” (The input is an angle in radians on the unit circle. The output is either the \(x\)- or \(y\)-coordinate of the point on the unit circle associated with the angle.) Since each angle only has one output associated with it, which we can visualize by drawing a line using the angle from the origin intersecting the unit circle at a single point, cosine and sine are functions.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion for the last question. After each student shares their response, provide the class with the following sentence frames to help them respond: “I agree because . . .” or “I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. This will help students make sense of the graphs of cosine and sine.
Design Principle(s): Support sense-making

9.3: Graphs of Cosine and Sine (15 minutes)

Activity

Now that the functions \(y = \cos(\theta)\) and \(y = \sin(\theta)\) have been introduced, the goal of this activity is for students to practice graphing these functions using technology. This informal exploration of the graphs gives students opportunities to make connections to their work with the unit circle, the Pythagorean Identity, and to an earlier unit in which they combined function types using arithmetic operations.

Monitor for students who use clear reasoning for the values of \(\theta\) where \(\cos(\theta)=\sin(\theta)\) to share during the whole-class discussion—for example, by thinking about the points on the unit circle where \(y=x\) or by recalling earlier work with tangent.

Launch

Arrange students in groups of 2. Tell students to first make their own prediction either in writing or a sketch, then share with their partner, and then check the predictions using graphing technology. Provide students with access to graphing technology.

Conversing: MLR2 Collect and Display. As students discuss their predictions, listen for and collect vocabulary, gestures, and diagrams students use to describe the graphs of cosine and sine functions. Capture student language that reflects a variety of ways to describe the similarities and differences between the functions. Write the students’ words on a visual display for all to see and remind students to borrow language from the display as needed. This will help students read and use mathematical language during partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. After students share their predictions with their partner, give students an opportunity to revise their predictions before checking them with technology. Display sentence frames to support students when they explain their prediction. For example, “One thing that is the same is . . .”, “One thing that is different is . . .”, “I noticed _____ so I . . .”, “Why did you . . . ?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

  1. Looking at the graphs of \(y=\cos(\theta)\) and \(y=\sin(\theta)\), at what values of \(\theta\) do \(\cos(\theta)=\sin(\theta)\)? Where on the unit circle do these points correspond to?
  2. For each of these equations, first predict what the graph looks like, and then check your prediction using technology.
    1. \(y=\cos(\theta)+\sin(\theta)\)
    2. \(y=\cos^2(\theta)\)
    3. \(y=\sin^2(\theta)\)
    4. \(y=\cos^2(\theta)+\sin^2(\theta)\)

Student Response

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

Student Facing

Are you ready for more?

For the equation given, predict what the graph looks like, and then check your prediction using technology: \(y=\theta+\cos(\theta)\).

Student Response

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

Anticipated Misconceptions

Some students may think \(y=\cos^2(\theta)\) or \(y=\sin^2(\theta)\) has special meaning and may be unsure how to make a prediction about their graphs. Ask these students how they would write the expressions on the left without using the square (\(\cos(\theta) \boldcdot \cos(\theta)\) and \(\sin(\theta)\boldcdot \sin(\theta)\)).

Activity Synthesis

Select previously identified students to share how they identified \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\) as the two inputs that make \(\cos(\theta)=\sin(\theta)\) true. If possible, display any student sketches of the unit circle for all to see.

Invite students to share how their predictions matched up to their partner’s and the actual graphs. Some important takeaways from this discussion are:

  • The cosine and sine functions are just like other functions we have studied. You can graph them and combine them with other functions using basic operations, like addition and multiplication.
  • If you square cosine or sine, the output values are all positive or 0.
  • The graph of \(y=\cos^2(\theta)+\sin^2(\theta)\) looks just like the graph of \(y=1\) and is a visual of the Pythagorean Identity, which states that \(\cos^2(\theta)+\sin^2(\theta)=1\) is always true.

During the graphing process, students may take note that the graph shows values for cosine and sine beyond inputs from 0 to \(2\pi\). If they question how this is possible and time allows, invite students to suggest the meaning of these inputs. Thinking about inputs outside of 0 to \(2\pi\) is the focus of the next lesson.

Lesson Synthesis

Lesson Synthesis

Tell students to close their books or devices and then display this graph for all to see:

Graph of a trigonometric function

Tell students that this is a graph of \(y=\sin(\theta)\). Here are some questions for discussion:

  • “What is the value of the second tick mark on the \(\theta\)-axis?” (\(\frac{\pi}{2}\) because this is where the sine function takes its maximum value of 1.)
  • “What is the first tick mark?” (\(\frac{\pi}{4}\) because it is halfway between 0 and \(\frac{\pi}{2}\).)
  • “What is the last tick mark?” (\(2\pi\) because that is a full circle and when sine completes its cycle of values. \(2\pi\) is the period of the sine function.)
  • “Why does the graph cross the axis at the 4th tick mark?” (That crossing corresponds to \(\pi\) on the unit circle, which is when the height of the point on the circle goes from above the \(x\)-axis to below the \(x\)-axis as \(\theta\) increases.)

9.4: Cool-down - Which Wave is It? (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Using the unit circle, we can make sense of \(\cos(\theta)\) and \(\sin(\theta)\) for any angle measure \(\theta\) between 0 and \(2\pi\) radians. For an angle \(\theta\) starting at the positive \(x\)-axis, there is a point \(C\) where the terminal ray of the angle intersects the unit circle. The coordinates of that point are \((\cos(\theta),\sin(\theta))\).

A circle with center, point B, at the origin of an x y plane.

But what if we wanted to think about just the horizontal position of point \(C\) as \(\theta\) goes from 0 to \(2\pi\)? The horizontal location is defined by the \(x\)-coordinate, which is \(\cos(\theta)\). If we graph \(y=\cos(\theta)\), we get:

Cosine graph. 

This graph is 1 when \(\theta\) is 0 because the \(x\)-coordinate of the point at 0 radians on the unit circle is \((1,0)\). The graph then decreases to -1 (the smallest \(x\)-value on the unit circle) before increasing back to 1.

We can do the same for the \(y\)-coordinate of a point on the unit circle by graphing \(y=\sin(\theta)\):

Graph. 

This graph is 0 when \(\theta\) is 0, increases to 1 (the greatest \(y\)-value on the unit circle), then decreases to -1 before returning to 0.