During this unit. I like to stage the progression of difficulty. I begin by asking students to verify identities that can be simplified using the fundamental identities. As they gain confidence, I add problems that require more algebra techniques. Today's bell work introduces students to the use of factoring in verifying a trig identity.
I give students a couple of minutes to think about the problem. I usually have a student that will see that both sides of the equation are in the form of a difference of squares. Many students will not see the algebraic structure of the problem. Once students are able to factor the students will quickly see how one of the factors is a Pythagorean Identity and can then verify the identity.
I begin working with students by asking questions to guide their thinking. My questions include:
The second question confounds some students. I find that if students do not quickly recognize that the expression can be factored, they end up deep in the weeds. Some students who see the opportunity to factor may hesitate because of the trigonometric functions. To help students see the underlying structure, I will model algebraic substitution as shown on the right side of page 2 of even odd bell. This simplification helps most of my students. I encourage them to recognize this and consider it as a useful problem solving tool: use algebra to change a difficult problem to a simpler problem you can can solve. After we factor the algebra expression I go back to the original identity and factor that expression. After completing the factoring the students realize that a factor is an identity and complete the verification.
Before concluding this section of the lesson, I will ask the class how this identity was different than the ones they had done previously. I let the student know that many identities require using algebra techniques learned in prior courses. These techniques include factoring, finding common denominators and multiplying by a conjugate.
In this section of the lesson, I will be using the even and odd functions to develop the sum and difference formula for sine, cosine and tangent. I always like to start by reviewing the general definitions of even and odd functions.
My class introduces the idea of even and odd trigonometric functions in Unit 4 (Trigonometric Functions as Real Valued Functions), but my students often find this concept very challenging. The activity in this lesson is designed to help students discover why cos(-x)=cos x and sin(-x)=-sin x. IN order to anchor the conversation in students' prior knowledge, I use a circle on the coordinate plane as a model. If students struggle with the meaning of the variables of x, y and r, I will work with specific points. I usually use (3,4) and (3, -4) because most of my students know the value of r when x = 3 and y = 4.
After about about 10 minutes I bring the class back together. We write the even odd identities on the board. I remind my students to add these identities to their reference sheet.
I ask the students "Why doesn't the value of cosine change?" On page 2 of even odd identities, I include the diagram that accompanied a student's response to this question. The student explained that the y-value is the only value that changes when we rotate in the opposite direction. This means that only the functions that involve y will have values that change sign. This is a great example of a student seeing structure and using it to reason.
I also want students to see how this works graphically. On page 3, I make a sketch of sine theta and cosine theta. I show the students how cosine is reflective about the y-axis. We compare this to y=x^2. I define this type of function as an even function. I develop a similar explanation for sine and how it is an odd function.
Once an identity has been developed students need to work with them. I begin the work by working with numbers first and then with verifying identities.
The first example makes the students find sin x by using the new identities. It is important for students to realize when we are find the values of an angle we are not wanting the trig value of the opposite angle. Page 2 shows how the students worked the problem.
After doing a finding values problem I give the students an identity to verify. Page 3 has the example I used. Students were able to verify the identity quickly since it did not require algebra techniques to verify.
Since the work in this unit is challenging and highly detailed, to end today's class I give the students time to work on problems from prior classes.
I also tell the students that we will be having a quick quiz over verifying identities. The students may use a reference sheet that has the identities but no examples can be on the sheet.