Pythagorean Identity

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Students will be able to use Pythagorean Identities to find the sine, cosine or tangent of an angle.

Big Idea

How can you use Pythagorean Theorem in two different ways to solve the same problem? Find out in this lesson.

Warm Up and Homework Review

10 minutes

I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up-Pythagorean Identities which asks students to determine whether to use sine or cosine to model a function.

I also use this time to correct and record the previous day's Homework.  


10 minutes

The first goal of this method is to refresh student knowledge of the Pythagorean Theorem.   Using the question “What is another thing besides trigonometry that we can do with right triangles?”

The next step is to build the Pythagorean Identities off of both the unit circle and the Pythagorean Theorem.  This can be run several ways depending on the level of the students.  For students who need more support, this connection can be built as a class.  For high achieving students, this can be worked on individually or in pairs first and then discussed as a class (Math Practice 1).  The key is that the students get the connection that sinΘ = y and cosΘ = x therefore, x2 + y2 = 1 is the same as cos2Θ + sin2Θ = 1. (Math Practice 7)

Finding Sine or Cosine Using the Pythagorean Identity

10 minutes

In the next portion of the lesson, I ask students to use this identity to find sine given cosine and vice versa.  It is helpful to note the other method of finding this is using Pythagorean Theorem to find the third side.  For higher achieving students, it will be a good idea to discuss the connections between these two methods. (Math Practice 7)  They both use a form of the Pythagorean Theorem, are they the same or different?  

Finding Tangent with the Pythagorean Identity

15 minutes

Our final goal is to find the tangent given either sine or cosine.  I give the students this problem as is without any warning about the change and let them struggle with it a bit (Math Practice 1).  There will be some that realize that you must find the other ratio using the Pythagorean Identity and use both to find the tangent.  I have one of these students share with the class.  If someone did it differently, they share as well.  Next, each student writes a summary of it in their notes. 

Once they have the new method, we’ll use it to solve a couple of problems.  

Exit Ticket + Homework

2 minutes

The first portion of this Homework practices using the Pythagorean Identity to find sine, cosine or tangent.  The final two questions assess student conceptual knowledge.  One question asks students to explain why it is unnecessary to use the Pythagorean Identity to find sine or cosine using tangent (Math Practice 3).  The final question asks students to explain the difference between the two uses the students now have to find trig ratios.  They are also asked which they prefer and why.

Exit Ticket

3 minutes

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.  

This Exit Ticket assesses students' ability to use the Pythagorean Identity for find a trig ratio.