This is the beginning of a unit where we formalize trigonometric identities that we have been using such as the reciprocal identities. The students will also analyze mathematical arguments (proofs) and develop mathematical arguments.
I quickly review the trigonometric ratios from the unit circle and the right triangle. I make a chart and put each trigonometric function in the chart. I have 2 columns that ask these questions?
I pick a student to write the class answers on the board. This student chooses other classmates to answer the questions. The only requirement is the "teacher" must pick a different student for each space.
After students have the "rules", students find values of the trigonometric functions when they know the value of one function. Students usually determine x=4 but forget -4. As they put up answers I say "How did you know x=4 not -4?" This leads to students determining there are 2 answers for most of the functions.
I like using students as the teacher when putting information on the board. My students love to use the Smart board and have control of the class. I have many students that participate in theater so having a student directing the class can be entertaining. I do at times need choose the "teacher" carefully since some of the students can be over dramatic and distract from the learning task.
This method works well if you have a student that is always wanting to answer the questions and the other students know they do not need to work since no one will call on them. Have the student who you know understands be the teacher and he can call on the other students.
One advantage of using a student to call on others and record the responses is that you can move around the room and see who is engaged and who is confused. If I see a student struggling I talk to the student to find out what they do not understand and then ask questions to have the "teacher" answer.
Before we get too far into this lesson, I want to make sure the students develop a common understanding of the term identities. As a start, I allow students to get out their phones to look up a definition. I explain that I want the mathematical definition. Most of my students use the site Math is Fun when I let them search in class.
Once students have searched, I ask a student to read the definition. I write the definition on the board so that we can discuss the definition. Some ideas that I want students to understand is that an identity is an equation and that it always true. When we discuss how an identity is an equation I talk about how there are 2 parts equivalent parts of an equation and that one side can replace the other side in another equation or expression.
After clarifying the definition I ask students to think of identities they know:
I put a quote on the board. I ask the following questions about this quote:
I give each student a piece of white paper. This sheet is for students to develop a reference sheet for the identities we will develop in this unit (see sample reference sheet). The students are told to title the page. I explain that this will be a sheet they use to remind them of important identities we will learn during the unit. Some students will not want to write the properties out. I try to explain that the sheet will help them remember properties once we get more. Even if they know these identities by having them on a sheet it will help jog there memory when they are working. I also allow this to be used sheet to be used on quizzes.
Students like a rational for new material. For this lesson I explain that I want to be able to determine the value of all the trigonometric functions without needing to use a triangle or using the parameters x, y, and r.
I put up the expression sin(theta)/cos(theta) on the board. Can we simplify this to a single trigonometric function? I let the students work on this for a couple of mintues.
When I see confusion I will clarify by saying:
Students will start to see what to do. Students work to simplify the expression. After 2 minutes I ask a student to share their process. (page 3)
Once the students see sin(theta)/cos(theta)=y/x students realize that y/x is tan(theta).
Can we now get a formula for cotangent in terms of sine and cosine?
Students are given another example to work. I tell them to use the identities on the reference sheet to find the other trigonometric functions.
Given sin(theta)=sqrt(5)/3 and cos(theta)=-2/3 find the value of the other trigonometric functions.
Students work on this and share their results. Students discuss how they found the values of each function. The question I usually start a discussion with is "what function value did you find first?"
The function that is the most difficult for the students is finding the value of tangent. The struggle is in simplifying the fractions.
One question that comes up a lot is why I write sec(theta) as -3/2 instead of 3/-2.
This question tells me that the students are still not proficient with fractions. To clarify I begin by asking what mathematical operation is done when you have a fraction. I then ask is -3 divided by 2 is equivalent to 3 divided by -2?
Once the students realize that both computations give the same decimal, I explain that the st standard convention is to either make the numerator negative or put the negative in front of the fraction. I usually put the negative on top to reduce confusion about what is written. I explain that the negative in the denominator is less noticeable and that the negative on the side sometimes looks like it is just part of the notation.
When I grade work I give credit for the negative in any position of the fraction and notice that the students will move the negative to the top as the year progresses.
Students need to practice a few problem over what was discussed today.
I assign p. 377, #14, 16, 22 from Larson's Precalculus with Limits and I give the students sometime to work. With about 5 minutes left in class I ask them to try and solve this problem in their groups.
Sin(theta)=-1 and cot(theta)=0 find the value of the other functions.
I collect the students' work on this task as an exit slip. This example is one that we have worked with before but the students may get confused when the find the value of tangent (theta). I want to see how the students deal with this issue.