Lesson 12

Tangent

12.1: Notice and Wonder: An Unusual Function (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that tangent is a function with similar features to cosine and sine. This will be useful when students describe features of the tangent function in a later activity. While students may notice and wonder many things about the table, the relationship between values of the three functions for different angles is the important discussion point.

This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1). Students will expand this table in the next activity.

Launch

Display the table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

\(\theta\) \(\cos(\theta)\) \(\sin(\theta)\) \(\tan(\theta)\)
\(\text-\frac{\pi}{2}\) 0 -1  
\(\text-\frac{\pi}{3}\) 0.5 -0.87  
\(\text-\frac{\pi}{6}\) 0.87 -0.5  
0 1 0  
\(\frac{\pi}{6}\) 0.87 0.5  
\(\frac{\pi}{3}\) 0.5 0.87  
\(\frac{\pi}{2}\) 0 1  

 

Student Response

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

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the graph. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the relationship between cosine, sine, and tangent does not come up during the conversation, ask students to discuss this idea. Highlight:

  • The value of tangent at \(\theta\) is \(\frac {\sin(\theta)}{\cos(\theta)}\).
  • There are angles where tangent doesn't exist.
  • Tangent can be positive or negative depending on the values of sine and cosine.

12.2: A Tangent Ratio (15 minutes)

Activity

In an earlier course, students learned to think of the tangent of an angle both as the ratio of sides in a right triangle and as the ratio of the sine and cosine of that angle. Earlier in this unit, students learned that one way to think about \(\tan(\theta)\) on the unit circle is the slope of the line connecting the origin and the point \((\cos(\theta),\sin(\theta))\) since \(\frac{\sin(\theta)}{\cos(\theta)}\) is the same as \(\frac{\text{rise}}{\text{run}}\).

The purpose of this activity is for students to develop their ideas of tangent as a function. They determine the value of tangent for specific points on the unit circle using the values of cosine and sine. Just as students extended the domain of sine and cosine to include any real number angle, they expand the domain of tangent. Unlike cosine and sine, however, tangent does not exist for all real numbers. Similar to the rational functions students worked with in an earlier unit, there are specific values which are not in the domain of tangent. In the following activity, students will identify these as the values where tangent has vertical asymptotes.

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

Launch

Arrange students in groups of 2. Depending on the level of challenge appropriate for students, this activity can be completed with or without a unit circle diagram or technology since students can use the values already given in the table and their knowledge of the symmetry of the unit circle to complete the table.

Engagement: Internalize Self Regulation. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite partners to form a group of 4. Students can divide up the work and share their findings with the group.
Supports accessibility for: Organization; Attention

Student Facing

  1. Complete the table. For each positive angle in the table, add the corresponding point and the segment between it and the origin to the unit circle.

    A circle with center at the origin of an x y plane. A point lies on the outside of the circle, in the first quadrant, labeled pi over 6.
    \(\theta\) \(\cos(\theta)\) \(\sin(\theta)\) \(\tan(\theta)\)
    \(\text-\frac{\pi}{2}\) 0 -1  
    \(\text-\frac{\pi}{3}\) 0.5 -0.87  
    \(\text-\frac{\pi}{6}\) 0.87 -0.5  
    0 1 0  
    \(\frac{\pi}{6}\) 0.87 0.5  
    \(\frac{\pi}{3}\) 0.5 0.87  
    \(\frac{\pi}{2}\) 0 1  
    \(\frac{2\pi}{3}\)      
    \(\frac{5\pi}{6}\)      
    \(\pi\)      
    \(\frac{7\pi}{6}\)      
    \(\frac{4\pi}{3}\)      
    \(\frac{3\pi}{2}\)      
    \(\frac{5\pi}{3}\)      
    \(\frac{11\pi}{6}\)      
    \(2\pi\)      
  2. How are the values of \(\tan(\theta)\) like the values of \(\cos(\theta)\) and \(\sin(\theta)\)? How are they different?

Student Response

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

Student Facing

Are you ready for more?

  1. Where does the line \(x=1\) intersect the line that passes through the origin and the point corresponding to the angle \(\frac{\pi}{6}\)?
  2. Where does the line \(x=1\) intersect the line that passes through the origin and the point corresponding to the angle \(\theta\)?
  3. Where do you think the name “tangent” of an angle comes from?
    unit circle centered at origin. angle the fraction pi over 6 drawn.

     

Student Response

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

Activity Synthesis

The goal of this discussion is for students to understand that while tangent is the ratio of sine to cosine, the tangent function has features not shared by the other two functions. In particular, there are angles where the tangent does not exist. In the next activity, students will consider what this means for the graph of tangent and if tangent is a periodic function.

Begin the discussion by inviting students to share their responses to how tangent is alike and different from cosine and sine. Here are some questions for discussion:

  • “Describe what happens to the value of tangent in the first quadrant as \(\theta\) increases.” (Tangent starts at a value of 0 and increases until at \(\frac{\pi}{2}\) the tangent does not exist.)
  • “How are the segments you drew related to the value of tangent—in particular, where tangent is undefined?” (Since the tangent of an angle is defined as \(\frac{\sin(\theta)}{\cos(\theta)}\), which is the same as \(\frac{\text{rise}}{\text{run}}\), the slope of the segment for an angle is the same as the value of the tangent of that angle. Since slope is undefined for vertical lines, which happens at \(\frac{\pi}{2}\)\(\frac{3\pi}{2}\), and every \(\pi\) radians after that, tangent is also undefined at these angles.)
  • “What are some negative radian angles where tangent is undefined?” (\(\text-\frac{\pi}{2}\)\(\text-\frac{3\pi}{2}\))
Conversing: MLR2 Collect and Display. During the synthesis, listen for and collect language students use to share their observations about the tangent function. Write the students’ words and phrases on a visual display and refer back to it throughout the lesson. Amplify words and phrases such as “the ratio of sine to cosine,” “undefined or doesn’t exist,” and “symmetry.” Remind students to borrow language from the display as needed. This will help students understand key features of the tangent function.
Design Principle(s): Maximize meta-awareness; Support sense-making

12.3: The Tangent Function (15 minutes)

Activity

The goal of this activity is for students to reason about what must be true about the tangent function and its graph before seeing the graph in the Activity Synthesis. The work here builds directly from the work with the table of tangent values in the previous activity. There are many important features of tangent and its graph, but the two main ones addressed here are its periodicity and asymptotes.

Launch

Arrange students in groups of 2. Since a goal of the activity is to draw conclusions about the graph of tangent, students should not use graphing technology during this activity.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “We will need to know . . .”, “We already know . . .”, and “_____ reminds me of _____ because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

Before we graph \(y= \tan(\theta)\), let’s figure out some things that must be true.

  1. Explain why the graph of \(\tan(\theta)\) has a vertical asymptote at \(x = \frac{\pi}{2}\).
  2. Does the graph of \(\tan(\theta)\) have other vertical asymptotes? Explain how you know.
  3. For which values of \(\theta\) is \(\tan(\theta)\) zero? For which values of \(\theta\) is \(\tan(\theta)\) one? Explain how you know.
  4. Is the graph of \(\tan(\theta)\) periodic? Explain how you know.

Student Response

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

Anticipated Misconceptions

Some students may not recall the cause of vertical asymptotes. Remind them of their work with rational functions, such as \(y=\frac{1}{x-3}\). Ask them, “What values of \(x\) make the denominator of the function equal to 0?” (\(x=3\)) “What happens to the function when the value of \(x\) is very close to 3?” (The value of the denominator is very close to 0, so the value of the function is getting very large.)

Activity Synthesis

The purpose of this discussion is to sketch a graph of tangent using what students have learned about tangent from the activity. Begin by displaying a blank graph showing \(\theta\) from \(\text-2\pi\) to \(2\pi\) on the horizontal axis and \(y\) values from -2 to 2 on the vertical axis.

Ask students,

  • “Where are the vertical asymptotes?” (\(\pm \frac{\pi}{2}\), \(\pm \frac{3\pi}{2}\), and so on.)
  • “Where are the zeroes of the function?” (At the multiples of \(\pi\).)
  • “Where is the function positive?” (Between 0 and \(\frac{\pi}{2}\), which is quadrant 1 on the unit circle, and between \(\pi\) and \(\frac{3\pi}{2}\), which is in quadrant 3 of the unit circle. Also any other positive or negative angles corresponding to points in those quadrants of the unit circle.)
  • “Thinking in terms of the unit circle, why is tangent positive in quadrants 1 and 3?” (These quadrants are where lines of positive slope that go through the origin intersect the unit circle.)

After asking each question, add on dashed lines for the vertical asymptote, points for the zeros, and color the axis to show where tangent is positive and where it is negative, respectively. Complete the graph or, if possible, invite students to use technology to graph the tangent.

Conclude the discussion by asking students to explain why the period of the tangent function is \(\pi\) instead of \(2\pi\), as it is for cosine and sine. (Each point on the unit circle has a point with opposite coordinates on the opposite side of the unit circle. These two points have the same tangent value, which means that after a half-circle of rotation, \(\pi\), the values of tangent begin to repeat.) Give students quiet work time and then time to share their work with a partner. Select 2–3 students to share their reasoning and any diagrams they used with the class.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their reasoning about the vertical asymptotes of the graph of tangent, present an incorrect response and explanation. For example, “To find vertical asymptotes, I just need to look at the unit circle where \(\cos(\theta)\) or \(\sin(\theta)\) equals to 0 because a fraction cannot have 0 in it.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who specifically reference the denominator in the fraction \(\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}\) and explain what happens when it gets close to 0. This helps students evaluate, and improve on, the written mathematical arguments of others, as they explain how to find vertical asymptotes on the tangent function.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

The purpose of this discussion is to highlight some features of the graph of tangent using the graphs of cosine and sine. Display this image for all to see:

Graph of 3 trigonometric functions

Begin the discussion by asking students to identify which graph goes with which function. Here are some questions for discussion:

  • “For which values of \(\theta\) does \(\tan(\theta)\) take the value 0?” (0, \(\pm \pi, \pm 2\pi\), and so on, because that is where sine has a value of 0.)
  • “Are there values of \(\theta\) where \(\tan(\theta)\) is not defined?” (Yes, whenever \(\cos(\theta) = 0\).)
  • “For which values of \(\theta\) is \(\tan(\theta)\) not defined?” (\(x = \pm \frac{\pi}{2}, \pm \frac{3\pi}{2}\), and so on, since this is where \(\tan(\theta)=\frac{1}{0}\).)
  • “What does the graph of \(y = \tan(\theta)\) look like near these values of \(x\)?” (This is where tangent has vertical asymptotes.)

Highlight that the features of \(\tan(\theta)\) (zeroes, asymptotes) repeat any time a multiple of \(\pi\) is added to the input \(\theta\). This is seen in the repeating (or periodic) shape of the graph of \(y = \tan(\theta)\).

12.4: Cool-down - The Graph of Tangent (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

The tangent of an angle \(\theta\), \(\tan(\theta)\), is the quotient of the sine and cosine: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Here is a graph of \(y = \tan(\theta)\).

A graph. 

We can see from the graph that \(\tan(\theta) = 0\) when \(\theta\) is \(\text-2\pi, \text-\pi, 0, \pi, \text{or } 2\pi\). This makes sense because the sine is 0 for these values of \(\theta\). Since sine and cosine are never 0 at the same \(\theta\), we can say that tangent has a value of 0 whenever sine has a value of 0.

We can also see the asymptotes of tangent \(\text-\frac{3\pi}{2}, \text-\frac{\pi}{2}, \frac{\pi}{2}, \text{and }\frac{3\pi}{2}\). Let’s look more closely at what happens when \(\theta = \frac{\pi}{2}\). We have \(\sin \frac{\pi}{2} = 1\) and \(\cos \frac{\pi}{2} = 0\). This means \(\tan \left(\frac{\pi}{2}\right) = \frac{1}{0}\), which is not defined. Whenever \(\cos(\theta) = 0\), the tangent is not defined and has a vertical asymptote.

Like the sine and cosine functions, the tangent function is periodic. This makes sense because it is defined using sine and cosine. The period of tangent is only \(\pi\) while the period of sine and cosine is \(2\pi\).