# Here's Proof

## Objective

SWBAT understand how to verify and apply the fundamental Pythagorean trigonometric identities.

#### Big Idea

Pythagoras again!? Make everybody happy â give your students another chance to connect trig functions to something they already understand.

10 minutes

## Put it Into Action

40 minutes

PROOF

• Beginning with the 2 equations on the board, I walk my students through a proof of the trig identity, with the requirement that they copy the steps as we go through.  I stop periodically to ask or answer questions.  You can see a video of a portion of this proof in my resources entitled here's proof narrative and the entire written portion as an educreations lesson entitled "Trig Identity Proof".  I didn’t video the entire exercise because it takes over ten minutes with my questions to the students, their responses, and their questions of me.  I ask questions while I demonstrate the verification on the board.  For a question like, “What part of the Pythagorean Theorem does the sin2 Θ correspond to?” I expect a simple response, but for more in-depth questions like "What happens to the denominator in the sin2 Θ + cos2 Θ expression?” I expect a more detailed explanation of why we can cross these denominators out (MP1).
• For the second part of this section I tell my students that they will each be working out a proof for two additional Pythagorean trigonometric identities, and that they need to complete these independently (MP2).  I also ask them to write out an explanation of the steps they take for each proof and why, to help them build their ability to communicate mathematically (MP3).  I post the two identities on the board then walk around as my students work on these, offering support, redirection and encouragement as needed.  The most common stumbling block for my students is being unsure of where to begin.  Since this is a challenge for them in many mathematical situations, I don’t generally tell them what to do, but instead try to help them choose by asking them questions like “What do you know about the tan Θ?” or “How are the cotangent and cosecant related?”  Some students still want more direction, but for the most part I try to refrain from staying with any one student for more than a minute (which is actually a long time in terms of total class time!).
• As most students finish this activity, I have them pass these papers in so I can look them over.  I don’t usually collect this kind of work and instead use it to initiate class discussion, but I feel that these papers give me insight into how well my students are making the connections between what they already know – Pythagorean – and what we’re learning – trigonometry.  I then ask if anyone can see any reason to understand or use these identities.  This question is usually met with blank stares or a response about needing to understand it for a later class or college.  I then work an example on the board showing how an understanding of the identities can help them solve for trig values.  The example I use is: “given sin Θ = 4/5 for π/2 < Θ < π, find the values of the other five trigonometric functions of Θ”.  I require my students to copy this problem as I work it on the board so they have an example in front of them for their homework.

## Wrap Up + Homework

5 minutes

When I’ve completed the example and answered any questions from my students I tell them that for their homework, they will be practicing applying what they’ve just learned.  I assign problems that specifically work with the Pythagorean Identities and finding values.  I tell students that I need to see their mathematics, but that I also need them to write about the problems to explain why they chose to solve each one as they did. (MP2)  A copy of the problems assigned is in my resources as Homework and the second page is an answer key.