ID# 3: Unit Q Concept 1: Pythagorean Identities

Inquiry Activity Summary

1.) Where does where sin2x+cos2x=1 come from to begin with (think Unit Circle!).

Well to answer this question we first have to think about the Pythagorean Theorem. It's an identity (which means it's a proven formula hat's always true) and we used it a lot when dealing with the Unit Circle. The variables we used were x (later we'll use x in the final form but in that case x stands for the angle), y, and r; so the formula was x^2 + y^2 = r^2. (I'll explain why the Unit Circle is important in a bit.)

 

Let's say we wanted the equation to equal 1. The easiest way to make that happen is to divide everything by r^2, which would leave us with x^2/r^2 + y^2/r^2 = 1.

 

Coincidentally, the cosine and sine ratios for the Unit Circle are x/r and y/r. Huh. Scratch that, not so coincidental.

 

If we replaced the ratios with sine and cosine (and switched the positions of sine and cosine but that's okay because it's addition), we'd get sin^2x + cos^2x = 1, which is the Pythagorean Identity!

 

2.) Show and explain how to derive the two remaining Pythagorean Identities from sin2x+cos2x=1.  Be sure to show step by step.
 

The remaining two Pythagorean Identities are 1 + tan^2x = sec^2x and 1 + cot^2x = csc^2x.

 

We'll start with the Pythagorean Theorem again: x^2 + y^2 = r^2. Instead of dividing by r^2, let's divide by x^2 and make the x term equal to 1. When we do that, we end up with  1 + y^2/x^2 = r^2/x^2. In the Unit Circle, y/x is the ratio for tangent and r/x is the ratio for secant. Replace them and you end up with 1 + tan^2x = sec^2x- another Pythagorean Identity!

 

Two down, we've got one to go. Start with the Pythagorean Theorem again: x^2 + y^2 = r^2. We've already used r^2 and x^2, so let's go with our last variable squared y^2. When you divide everything by y^2, you get x^2/y^2 + 1 = r^2/y^2. In the Unit Circle, x/y is the ratio for cotangent and r/y is the ratio for cosecant. Replace them (and switch the x and y terms around if you like) and you end up with 1 + cot^2x = csc^2x- the last Pythagorean Identity.

 

Inquiry Activity Reflection

1.) “The connections that I see between Units N, O, P, and Q so far are…”

 

The connections I see between the three units are the use of the Unit Circle and the use of trigonometric ratios.

 

“If I had to describe trigonometry in THREE words, they would be…”
 

The three words would be "memorization", "ratios", and "angles".