I/D3: Unit Q - Pythagorean Identities

Inquiry Activity Summary: 
1. sin2x+cos2x=1 
This is a Pythagorean Identity (a proven fact/formula that is always true), so, we will be using the Pythagorean Theorem in order to begin deriving it. Originally, our Pythagorean Theorem is a^2 + b^2 = c^2. However, we learned in recent units that we deal with triangles on graphs, specifically when we are dealing with the Unit Circle. The same situation is still present, therefore we want to adjust our formula. The legs of your triangle are measured in how much you go left/right on the graph (x-axis) and how much you go up/down (y-axis). We also remember that our hypotenuse of our triangle is also the radius of the Unit Circle. So we go from a^2 + b^2 = c^2, to x^2 + y^2 = r^2. Now that we have our "new" Pythagorean Theorem, we continue on and think of how on earth we simplify it into our Pythagorean Identity.
Our formula is  x^2 + y^2 = r^2. We want to get to sin2x+cos2x=1. To do that, we divide everything by r^2. If we divide everything by r^2, we get this: x^2/r^2 + y^2/r^2 = 1 (r^2/r^2). Already, we can tell that r^2 divided by r^2 is 1. Now we can look at the other parts; x^2/r^2 is basically (x/r)^2 and y^2/r^2 is (y/r)^2. Immediately, you should notice something about these variables and RATIOS. The ratio for cosine is x/r. The ratio for sine is y/r. *AHA MOMENT*
Now that we are aware of this relationship, let's plug them into our formula. So, (x/r)^2 becomes cos^2x and (y/r)^2 becomes sin^2x. Finally, we end up with cos^2x + sin^2x = 1.

This picture is clarification of what I described in the text ABOVE. 

Don't believe that it actually works? We can use any of the "Magic 3" ordered pairs from the Unit Circle to show that it is true. Look in the picture BELOW to see how I used the 60 degrees ordered pair in order to prove this identity true.

Simple matter of plugging in numbers into our identity and verifying. 

2. Other Pythagorean Identities 

Now that we have our main Pythagorean Identity, we will use it to find our other two. 

• The first Pythagorean Identity contains secant and tangent. The only way to get that is by dividing by either cosine or sine. For this one, we want to divide everything by cosine because the ratios will simplify into what we want: tangent and secant. We divide sin2x by cos2x and we get tan2x. How do I know? Besides from memorizing this identity, we can use logic. Sine's ratio is y/r and it is being divided by cosine's ratio, which is x/r. Another way we can look at that is y/r TIMES r/x. So as a result we have y/x which is tangent. Cosine divided by cosine is simply 1. And 1 divided by cos2x is sec2x (because secx = 1/cosx; we just powered up because everything is being multiplied in this reciprocal identity). The picture below gives clarification of what you should be left with.

This is what our Pythagorean Identity is. 

• The other Pythagorean Identity contains cosecant and cotangent. Because we divided the last identity by cosine, we want to divide by sine this time. We divide sin2x by sin2x and get 1. We divide cos2x by sin2x and we get cot2x. (x/r TIMES r/y = x/y). We divide 1 by sin2x and we end up with csc2x. The picture below will have the Pythagorean Identity and work along with it. 

This is what our 3rd Pythagorean Identity looks like. 

Inquiry Activity Reflection: 
1. “The connections that I see between Units N, O, P, and Q so far are…” the continual use of trig functions (and the things we are able to use with them), and the references towards Unit Circles and the triangles within it.
2. “If I had to describe trigonometry in THREE words, they would be…” Ratios. Are. Everywhere.