Define the secant, cosecant, and cotangent functions as the reciprocal of the cosine, sine, and tangent functions, respectively.
Understand how the zeros, vertical asymptotes, and range are related for a trigonometric function and its reciprocal function.
Quick Lesson Plan
Lesson 7.4 Applet: Geogebra Applet
Today we’ll use another Geogebra applet to study the reciprocal trig functions! First, students look at a static image of the applet and do a notice and wonder. If you can’t print in color, we suggest providing students with colored pencils so they can color the segments in the diagram appropriately.
We do not expect that they have ever heard of the reciprocal functions yet, so do not assume they know that csc means cosecant, or that sec means secant. Question 2 starts to build their intuition around the relationships between the six trig functions by having them predict how the lengths of the segments change as theta increases slightly (still in the first quadrant!).
Students will then use the applet to do the rest of the activity. Note: we have budgeted two days for this lesson because there is a lot of content! You have some options for how you can divide the lesson. We suggest having students work on questions 1-8 on day 1 and then debriefing those questions. On day 2, students can do the rest of the activity, the QuickNotes, and the Check Your Understanding. This should give them plenty of time to really explore! There are plenty of additional things to notice in the applet besides the topics specifically referenced in the questions.
It is critical that students recognize the reciprocal relationship between the trig functions. We suggest having groups check in with you before moving on to question 4 so you can assess what they took away from question 3. Use guiding questions to get them to arrive at the reciprocal relationship without giving it away (see Monitoring Questions below). You may help them pause on a value where the reciprocal relationship is easier to see, such as when the sine is ½ and the cosecant is 2.
When students begin to graph, they will likely start by graphing the vertical asymptotes and points where the output is 1. Although we do not provide a table for students to fill out, feel free to mention this as a strategy if students are totally unsure how to sketch the graph. The Geogebra applet gives them plenty of values to work with!
As the sine value gets bigger, does the cosecant value get bigger or smaller?
Hmmm, when sine is 1, the cosecant is also 1. What does that mean?
Why is the cosecant value never between 0 and 1? Why is the sine value never greater than 1?
Will Point J or Point C ever be within the circle? Why or why not?
Why is the cotangent function decreasing?
What transformations occurred to get from the graph of the tangent function to the graph of the cotangent function?
(Bonus!) What do you notice about the trig functions that appear in ∆ACO compared to the trig functions that appear in ∆JAO? (Cosine, cosecant, and cotangent are the sine, secant, and tangent of the COmplimentary angle in the triangle (angle JOA).)
A key understanding from this lesson is that knowing how the sine and cosine and tangent functions behave is enough to determine the behavior of their reciprocal functions. We don’t want students to think that they have to memorize three completely new functions! Continue to emphasize when a rational expression is equal to 0 and when it is undefined.
One downside of the applet is that the angles are measured in degrees rather than radians. This is because when the angle is given in radians and NOT in terms of π, students (rightly) have a very difficult time understanding the angle measure. Because all the key features occur at multiples of π/2, switching to a graph where the x-axis represents the angle in radians will hopefully not be too much of a jump. You could ask students to give their answers in the activity as radian measures throughout.
Note that graphing and analyzing transformations of the reciprocal trig functions is not officially part of the AP Precalculus course framework. However, we chose to ask question 3 of the Check Your Understanding to see if students could apply previous learning to new parent functions. Any homework or additional practice you assign should not be heavy on graphing transformed reciprocal trig functions (unless of course it’s important to you–not everything is about the AP Exam!)