Proving Identities

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SWBAT use fundamental trigonometric identities to prove other identities.

Big Idea

How can you use the fundamental identities show 2 expressions are equivalent?

Bell Work

15 minutes

Here are the slides for today's lesson. If you use SMART notebook, I have attached the notebook file as well.

As students enter I put the 3 homework problems on the board (Slide 1).  Once we settle I have different students share their solutions with the class.

I ask the following questions:

  • Do you have any questions about the process?
  • Can you explain the second, third, etc. step to me?
  • Did anyone do the problem a different way?

As we look at the work students begin seeing different ways to do the problems. Page 2 and 3 of the student work show other ways the students worked the first problem. For the students, having classmates demonstrate different methods will help them increase their understanding. If one method is confusing another one may be more clear to them.

Proving Identities

25 minutes

We have looked at several identities that I consider fundamental or basic identities.  The students have  been using these to simplify expressions. I now ask the class "Wouldn't it be nice to know what you want the answer to look like when you simplify?"

Students are usually thrilled to know the answer.  I usually make a comment like If you know the beginning and the end of a problem what do you think you will need to do? Students are generally quick to say, "fill in the middle."

Now, I give the students an identity to solve. I ask, "Which side looks more complicated?" I will then tell the class to start with the complicated side (in this case the left side) and try to make it become the other side. Try not to do anything with the right side in this case. That is the traditional method of verifying an identity.

I give the students a minute to discuss what we might do with this problem with the students in their group. After the quick discussion I will ask students questions like these:

  • Why did you use sin x/tan x instead of 1/cot x?
  • Why did you decide to work with that side first?

On the third slide a student decided to use the inverse property for tangent. The student became stuck. I had the student show his work and I then ask the class to explain what he could do to finish. In this case the student got to cos x/cot x. The students told him to use the identity that cot x=cos x/sin x. We put the remainder of the problem on the board for everyone to see.

This video shows a student demonstrating his reasoning for the example.  

I give the student another  example. Students work with their groups to try and verify the identity. 

This time I have the students try the problem in their groups. As the students work I move around the room and answer questions or quiz students on errors. I may ask why did you put that statement down. Where did you get this step.  After a 2-3 minutes I have a student share their process.  If students have done this in different ways we will put the different processes up and discuss whether the processes work.

On the first slide a student got stuck so the class worked to help the student.  Some students immediately knew that 1-cos^2 x=sin^2 x. The student that volunteered decided to replace 1 then combine like terms.  For students that have recognized how to rearrange the Pythagorean identities this method is what they will do.  We discuss how it is the same as just finishing by replacing 1=cos^2 x with sin ^2 x.

I wanted to show students how they can also solve this by adding fractions. Slide 2 gives students this process.  As you can see I had to remind students how to add fractions with numbers because the trigonometric functions confused them.  I reminded students that we will be doing a lot of algebraic methods during this unit.

For this day we only did 3 examples.  I had more ready but the students spent a lot of time analyzing work and asking questions about students reasoning.  


5 minutes

With a few minutes left in class give the students three problems from Larson's Precalculus with Limits (page  385, problems 10,11,12). I allow students a few minutes to discuss the work with others. I have chosen problems that I expect all students can do on their own.  These three problems do not require many algebra techniques, which is where most students struggle. As we continue adding new identities we will move to more complex identities to verify.