I/D3: Unit Q - Pythagorean Identities

Inquiry Activity Summary

The Pythagorean is here again. We first saw it in unit N in the unit circle, and it relates to trig identities as well. Using x, y, and r, we can show that. (x^2/r^2)+(y^2/r^2)=(r^2/r^2) which is also (x^2/r^2)+(y^2/r^2)=1. This can be simplified to (x/r)^2+(y/r)^2=1. sin2x+cos2x=1 comes from the unit circle. The ratio for cosine on the unit circle is x/r. The ratio for since is y/r. Here we are going to take one of the magic 3 ordered pairs: 30 degrees. cos30 is equal to radical 3 over 2 and sin30 is equal to 1/2. Radical 3 over 2 squared is equal to 3/4. 1/2 squared is equal to 1/4. 3/4+1/4=1 which is where you get the one on the right side. Since we squared these fractions, we have to show it in our identities also. This identity is actually the Pythagorean theorem moved around, hence it is also called the Pythagorean identity. Keep in mind that an identity is "a proven fact and formula that is always true." This tactic works with any degree from the magic 3.

We can also derive the secant and tangent from sin2x+cos2x=1. If you divide all of that by cos2x, you will end up with tan2x+1=sec2x. With tanx=sinx/cosx, multiply the left and right sides by themselves to get a squared value for both. tanx*tanx=(sinx/cosx)(sinx/cos). In the end you will get tan2x=sin2x/cos2x.

Inquiry Activity Reflection

1. The connections that I see between Units N, O, P, and Q so far are that they all relate to triangle and they all use the six trig functions.
2. If I had to describe trigonometry in THREE words, they would be challenging, connected, and unexceptional.

I/D2: Unit O - Derive the SRTs

INQUIRY ACTIVITY SUMMARY
As seen in the square, you have to cut it diagonally because that is how you can get two triangles. If they are equally bisected, they will form two 45-45-90 triangles. You get the hypotenuse by using the Pythagorean theorem (a^2+b^2=c^2). We are told that the side lengths of the square are 1. That means the side lengths of a and b of the triangle are 1 as well. You take (1)^2+(1)^2=c^2. That will give you c=radical 2; this is your hypotenuse. "n" means the variable used to find your sides. Since in a 45-45-90 degree triangle the sides corresponding to the 45 degree angles are the same, we can label them with "n." 





In the 30-60-90 triangle, I got it by bisecting an equilateral triangle down the center. Since the sides of the equilateral are equal to 1, I knew the hypotenuse was 1. The base is 1/2 since we cut the triangle in half. I then used Pythagorean theorem to find the height which gave me a=radical 3 over 2. This translates to the normal pattern because side a is n equal to n. The hypotenuse is double that. Here we see that side a (1/2) was multiplied by 2. (2)(1/2)=1; that is our hypotenuse. To find the height, we are most familiar with n radical 3. Our n in 1/2. We multiply that by radical 3. That gives us radical 3 over 2 as our height. We use n because that's the variable we are most familiar with when we first learned about triangles in geometry. In this problem the b is n, the hypotenuse is 2n and the height is n radical 3.

INQUIRY ACTIVITY REFLECTION

Something I never noticed before about special right triangles is
that they can be made from squares and triangles. After learning the unit circle, I thought special triangles were only used in that.
Being able to derive these patterns myself aids in my learning because
I now better understand my last unit as well as understand where my values come from. Before I just had to memorize the values of each side without knowing how I got them.

I/D1: Unit N - How do SRT and UC relate?

INQUIRY ACTIVITY SUMMARY
Describe the 30* triangle
Click HERE.
Describe the 45* triangle
Click HERE.
Describe the 60* triangle
Click HERE.


1. The coolest thing I learned from this activity was...
how to find these points and coordinates mathematically instead of memorizing it by heart.
2. This activity will help me in this unit because...
I can now better understand how to label my triangles and my unit circle. It also showed me patterns I didn't notice before.
3. Something I never realized before about special right triangles and the unit circle is…
that they actually relate to each other! I was told that in past math courses, but I didn't quite grasp what it meant.