Lesson 12

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.

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.

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.

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).

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.

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?” (\(\text-\frac{\pi}{2}\), \(\text-\frac{3\pi}{2}\))

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.

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.

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.)

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.