I/D#3: Unit Q - Pythagorean Identities Concept 1: Using fundamental identities to simplify and verify expression (simple, one or two step identities)

1. When you draw a triangle inside a Unit Circle we use Pythagorean Theorem to finding the missing sides. We can use the Pythagorean Theorem to show why it is an identity.

It is referred to a Pythagorean Identity because you are using the Pythagorean Theorem to solve for the triangle and we have our ratios at the end of the equations which show that it is sine and cosine.  

Now if you were to plug in any of the coordinate points from the 30 45 60 degrees reference angles, you will always get answer that is equal to 1. For example:

2. To get the second identity from the Pythagorean Identities you have to take what you got from the first identity which was sin^2x+cos^2x=1 and divide it all by cos^2x. From there you use your ratio and reciprocal identities to find what each divided problem is equal to. And from there you have your tangent and secant identity.

To get the third identity you once again take your first Pythagorean Identity which was sin^2x+cos^2x=1 and divide it all by sin^2x. You then have to multiple divided identities and you use your ratio and reciprocal identities to find what each one equals too and from there you have your final identity that uses cotangent and cosecant

I/D# 1: Unit N Concept 7 : Knowing All Degrees and Radians Around the Unit Circle, Knowing All the Ordered Pairs Around the Unit Circle, Understanding and Applying ASTC to the Unit Circle

30 Degrees Triangle 

45 Degrees Triangle 

60 Degrees Triangle & The Unit Circle

The coolest thing I learned from this activity was… the special right triangles that we learned about in geometry (which was way back in freshmen year) is actually related to the Unit Circles we began learning about sophomore year. I now know that the ordered pairs aren't just numbers put together randomly, I understand where these numbers come from.
This activity will help me in this unit because…I understand where these numbers come from when it comes down to labeling all ordered pairs. I am able to figure out how to get these ordered pairs if I ever forget what goes where. I can just draw my triangles and solve them to find my ordered pairs.
Something I never realized before about special right triangles and the unit circle is…that they are related. When I first started learning about special right triangles I thought it was going to be something that we just learned and weren't going to go back on it and use it later on like we do when we learned algebra. We are actually taking what we learned about special right triangles in geometry and finding the ordered pairs of them for the Unit Circles we are now learning about in Math Analysis (Pre-Calculus).