I/D #3

Inquiry Activity Summary
1.Today we will be looking at the devriation of the "identity". "Identity" is a proven facts and formula that is always proven true. For example, the pythagorean theoream of a^2+b^2=c^2 is an identity.

To begin with, the pythagorean theorem of a triangle using x, y, and r, is x^2+Y^2=r^2. To make it equal to 1, we would have to divide everything by r^2. Since the ratio for cosine on the  unit circle is x/r and the ratio for sine is y/r, we can say that cos^2theata+sine^2theata=1. Therefore, we can conclude that sin^2x+cos^2x=1 is a pythagorean identity because it is proven to be exactly the same as x/r^2+y/r^2=1. The replacement of x and y with cos and sin proves that it is true.

Another way that is proven true is to plug in the "magic 3" ordered pairs from the unit circle. When we plug in the pairs, we know that it is equal to 1. Therefore, it is an identity.

2. To derive the identity from sin^2+cos^2x=1, first divide everything by cos^2. Sin/tan is equal to tan. Because it is squared with an x, we leave that the same. Cos^2x/cos^2x is equal to 1. 1/cos^2x is equal to sec^2x

To derive it from cosecant and cotangent, you set up the same. Sin2^x+cos^2x=1. To begin with, you divide everything by sin^2x. Sin^2x/sin^2x=1. Cos^2x/sinx^2 is equal to cot^2x. 1/sin^2x is equal to csc^2x.

Inquiry Activity Reflection:

The connections that I see between Units N, O, P, and Q so far are unit circle has relation between these identities. These identities is basically derived from the pythagorean theorem. The pythagorean theorem can also derived from these triq functions.

If I had to describe trigonometry in THREE words, they would be identity, magic ordered pairs, and pythagorean theorem.