The Pythagorean Identities
Lesson 2 of 14
Objective: SWBAT prove the Pythagorean identities.
We begin today by doing a problem similar to the homework from yesterday. As the students work on the problem I walk around the room to see who is struggling and who has a good understanding of yesterday's ideas. If I see a student struggling I will question the student and fix any misconceptions.
The students share their responses for the class to review. I ask "How may of you used your reference sheet to help answer this questions? What formulas were did you use?"
I expect some students to struggle with the fact that cotangent is undefined. If this occurs I ask, "What will make a fraction undefined?" Many students will understand that dividing by zero gives an undefined result. "If dividing by zero means it is undefined how could we write the value of cotangent as a fraction?" Once this is established I will ask, "So if cotangent is undefined what is the value of tangent?"
I now ask if the students can determine the value of sine theta. If the students are having trouble I refer back to the reference sheet and ask if we have anything we might use. We have not learned the Pythagorean Identity so we have to use the identity of tan (theta)=sin(theta)/cos (theta). The second page of the Bell Work shows how the students used the property to find the value of sine.
Today, we are adding three more identities to the students' reference sheet which will give them more tools to use when evaluating or simplifying expressions.
We will develop the first Pythagorean Identity as a class. I begin by reminding students of the sine and cosine ratios when you know an ordered pair in the coordinate plane (Slides page 1). I ask the following questions:
- What is r and how is it determined? Many students need to have the right triangle drawn from the the point and the sides label as x, y and r. Once the triangle is drawn the students know that r is found by using the Pythagorean Theorem.
- How could we rewrite x^2+y^2=r^2 so it is equal to 1 instead of r^2? Students are given a minute to think.
Students determine we need to divide each side by r^2. We have an equation that can be rewritten as (x/r)^2 and (y/r)^2.
As this equation is written on the board I will hear some students make comments as like "oh x/r is cosine." When that ah-ha moment is occurring I ask students to explain what they are seeing. Once the student explains what they are noticing I ask if we can replace the x/r with cosine and y/r with sine.
I do not move too quickly with the notation because students get confused about (sin(theta))^2 and sin^2(theta).
After discussing the meaning of the expression, I show the students how to write a trigonometric function that is squared. On page 3 of the Slides I have written the square of the expression in three different ways. I discuss what each expression means with the class. I also tell students that they can write sine(theta) squared with parenthesis or in the traditional way.
Now that we have the first Pythagorean Identity I have the students label their reference sheet and write the identity on their reference sheet. I have the students write the property as sin^2(theta)+cos^2(theta)=1 and as (sin(theta))^2+(cos(theta))^2=1. The second notation will help students as they use the property when they only know sine or cosine.
Students now work a problem using the Pythagorean Identity.
sin(theta)=12/13 what is cos(theta)?
Students work for about a minute. I see what students found to be the answer. We share the process with the class (Example 1). When students work this example I ask students whether cosine is positive or negative. Students often forget that you have 2 answers when you take the square root as you solve.
Once we determine there are 2 different answers I ask "What do you need to know to determine the sign of cosine?" and What quadrant would the angle need to terminate to have a negative cosine value? a positive cosine value?"
Instead of just giving students the other identities I want to find the other identities by manipulating the first identity algebraically . I begin by writing the first Pythagorean Identity on the board (see Property 2). I then say, "Let's divide each term by sin^2(theta)."
As we work we again have to see how 1/sin^2(theta) is the same as (1/sin(theta))^2 are equivalent expressions. The work here is also the first steps in writing arguments for verifying identities.
As we work students should have their reference sheets out. I have some students that quickly see how we can write the equations but other students will need some questioning to see the process. I make a comment about having an identity for 1/sin(theta). I ask "since 1/sin(theta)=csc(theta) can we replace or substitute and rewrite the expression?" This can confuse some students because they only think can we can replace variables with numbers. In some classes we have a discussion about how to substitute an expression with an equivalent expression. I have students explain why this is possible and clarify by rephrasing what students say.
For the last identity I again put the original Pythagorean Identity on the board. I make a comment "we just divided each term by sin^2(theta) then replaced some terms with equivalent expressions to get a new identity. Could we get a new identity by dividing by cos^2(theta)?"
I have the students work in groups to find the new identity (Property 3). After a few minutes I pick a student to put a solution on the board. Before we move on, I ask my students put the 2 new identities on the reference sheet so that they now have 3 Pythagorean Identities.
Practice and Closure
Students need to practice with these identities. I give students the following problem:
sec u=-3/2 and tan u>0. Find the value of the other trigonometric functions.
Students are given a few minutes to work on the problem with their groups. As they work I note comments they are making to each other. Some students immediately know how to find cos u but are struggling finding the other values. Some are not sure what to do with the second part of the prompt.
I have students put results on the board and then begin asking questions:
- How did you know that cos u=-2/3?
- If we know sec u can we find tan u? What would you use to find tan u?
- How can you find sin u once you know cos u?
- If you know cos u and tan u how can we find sin u?
I determine the question by what the students were able to find on their own. If the class found all the parts I use the above questions to help students see different methods for find the trigonometric values.
At this point the class is getting near the end so I ask a question that is a preview of simplifying expressions. I say that there are 3 different expressions that can replace tan u. Find at least 2 of these expressions and put this on an exit slip.
Most students will write 1/cot u and sin u/cos u. A few students will give +/-sqrt(1+sec^2 u) as the third expression.