Lesson 9

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.

Display the image for all to see.

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

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.

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 \(\frac{3\pi}{2}\).

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.

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.

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.

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 \(\sin(\theta)\boldcdot \sin(\theta)\)).

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.