Experience First:

After having focused primarily on the sine and cosine functions in Unit 6, we now take a deep dive into the tangent function–and we have a great Geogebra applet to help! In this activity students explore how the slope of the terminal ray of an angle changes as the angle increases. 

 

First, students are reminded of how the sine and cosine values can be seen on the unit circle as the vertical distance and horizontal distance, respectively of a point on the circle on the terminal ray of an angle. They then connect all the different representations of the tangent of an angle in question 4.

 

Now for the fun part! Using the Geogebra applet, students will be able to change the value of theta using a slider. Note that we specifically use values of theta that are not just the “key angles” on the unit circle. This helps show students the continuity of the tangent function and gives a better picture of its overall behavior. Students will see that the slope of segment OA is positive in quadrants I and III and negative in quadrants II and IV from a graphical perspective (rather than analytically comparing the signs of the x and y coordinate). Students will also see that the slope at π/2, 3π/2, and -π/2 is undefined. The applet literally says that tan(theta)=?. This is a nice way for students to review that vertical lines have undefined slope!

 

Questions 5f and 5g tie back important ideas from Unit 1 about concavity. Students look at the rate at which the slopes of segment OA are changing over equal intervals to preview the idea that tan x is concave up for 0<x<π/2 and concave down for -π/2<x<π/2, but this idea won’t get solidified until question 11.

 

In question 6, students will notice that each tangent value repeats three times over the interval -π to 2π, meaning that the tangent function goes through 3 complete cycles over that interval. They use this to establish that the period of the tangent function is π. 

 

Stop and debrief page 1 before having students continue on to page 2.

 

On the second page of the activity students complete a table of values to help them sketch a graph of the tangent. The table might seem long, but many of the values are 0, 1, -1, or undefined. We chose one additional tangent value (±3.73) to give students a better idea of the behavior of the tangent function near the asymptotes. 

 

Question 8 reviews the idea of periodic behavior and questions 9-11 can be answered by looking at the graph. In question 11c, students connect graphical and analytical representations of concavity.

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Monitoring Questions:

• What does an “undefined” slope mean? What are the slopes right before the undefined slope? What are the slopes right after the undefined slope?

• How does knowing where the slope of OA is zero reveal about the graph of the tangent function?

• Are the slopes of OA increasing or decreasing as the value of theta increases?

• Is there an angle on the unit circle where the slope of OA is exactly 23? Exactly 142.1? How do you know?

• Why is the period of the tangent function shorter than the period of sine and cosine?

• How does the graph support your answer to 5a and 5b (where you identified QUADRANTS in which the tangent was positive and negative)?

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Formalize Later:

Be very careful with the language you use around the tangent function’s concavity. We generally talk about slopes of “the graph” but students are also talking about the slopes of segment OA which represent the VALUES of the tangent function, not its rate of change! Avoid saying “the slopes” and instead be specific with phrases like “the slope of segment OA” or “the values/outputs of the tangent function” or, when talking about the tangent function’s rate of change, “the slopes of the tangent function” or “the rate of change of the tangent function”.