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 the sine and cosine functions to include any real-number angle, they expand the domain of the tangent function. Unlike cosine and sine, however, tangent does not exist for all real numbers. Similar to the rational functions that students worked with in an earlier unit, there are specific values that are not in the domain of the tangent function. In the following activity, students will identify these as the values where the graph of the tangent function has vertical asymptotes.

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

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 tangent does not exist. In the next activity, students will consider what this means for the graph of the tangent function and if tangent is a periodic function.

Direct students' attention to the reference created using Collect and Display. Ask students to share their responses to how the tangent function is alike and different from the cosine and sine functions. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (For example, “For some angles, there was no value for tangent, or tangent was undefined.”) Here are some additional questions for discussion:

• “Describe what happens to the value of tangent in the first quadrant as increases.” (Tangent starts at a value of 0 and increases until at 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 , which is the same as , 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 , , and every radians after that, tangent is also undefined at these angles.)
• “What are some negative radian angles where tangent is undefined?” (, )

If students don’t remember the cause of vertical asymptotes, consider saying:

• “Tell me more about the value of tangent at .”

• “Consider the rational function . What happens to the graph of at ? How does that relate to our graph of tangent?”

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

Ask students,

• “Where are the vertical asymptotes?” (, , and so on.)
• “Where are the zeroes of the function?” (At the multiples of .)
• “Where is the function positive?” (Between 0 and , which is Quadrant I on the unit circle, and between and , which is in Quadrant III 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 I and III?” (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, add 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 function.

Conclude the discussion by asking students to explain why the period of the tangent function is instead of , as it is for the cosine and sine functions. (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, , 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 with the class their reasoning and any diagrams they used.