I/D3: Unit Q - Pythagorean Identities

I - Inquiry Activity Summary 

1) Before we dive too deep into Pythagorean Identities, it is important to know what an "identity" is. An identity is a proven fact or a formula that is always true. The Pythagorean Theorem is an identity because it is always true! In other words, "x²+y²"will always equal "r²." We can make this equal to one by dividing the entire equation by "r²." This action would yield, "(x/r)²+(y/r)²  = 1." By using these letters, we can relate the Pythagorean Theorem to the unit circle! "x/r" is equivalent to the trig ratio of cosine while "y/r" is the ration for sine. These ratios are created by setting the Pythagorean Theorem equal to one; thereby allowing us to use trig functions in relation. Not only do we achieve our trig functions, but we further our use of Unit Circle knowledge by noticing that the "1" that we set the Pythagorean Theorem to is also the radius of the Circle. From this we can conclude that the two are directly tied together! Since (x/r) is equal to cosine and (y/r) is equal to sine, we can substitute the two ratios for trig functions! In other words, we could change "(x/r)²+(y/r)²  = 1" to "cos²x + sin²x = 1."

This is referred to as the Pythagorean Identity because it is a proven formula that is always true; a formula which equates the Pythagorean Theorem. 

We can see how the values of the unit circle prove that the Pythagorean Identity is legitimate. In the photo above, I used 60 degrees which has a value of (1/2, rad3/2). If we plug this into the identity, square each value, and add them together, we prove that the formula is correct as it equals one.  

2)

a - in order to derive the identity involving Secant and Tangent, you need to divide by cos^2x! this will yield "1 + tan^2x = sec^2x." The ratios are very important in this step. By dividing by sine, you are dividing by ratios. This can lead to changes in ratios which then result in changes to the trig functions!  

b - We follow similar step to derive the identity involving Cosecant and Cotangent! By dividing we yet again yield different ratios and different trig functions resulting in our identity. 

II - Inquiry Activity Reflection 

“The connections that I see between Units N, O, P, and Q so far are…” the repeated references to the unit circle and even the trig functions.

“If I had to describe trigonometry in THREE words, they would be…” unit circles, ratios, and triangles.