I/D #3: Unit Q Concept 1- Using Fundamental Idenities to Simplify or Verify Expressions

1. An "identitiy" are proven facts and formulas that are ALWAYS true. The Pythagorean Theorem is a type of "identitiy" because we get proven facts and formulas that end up being true for any numbers we plug into it. With using the Pythagorean Theorem we use for this unit we use x, y, and r. We use these instead of a, b, and c because on a graph it when we graph it becomes x, y, and r.

As you see in the picture above all we had to do to get the Pythagorean Theorem equal to 1 is divide by r^2. Once we had r^2 it became equal to 1. From dividing by r^2 I noticed that we got x/r which is the ratio for cosine we used for the unit circle. I also noticed that we got y/r which is the ratio for sine we used for the unit circle. If we subsitiute those two for cos and sin we get cos^2x + sin^2x = 1. We can conclude that cos^2x + sin^2x = 1 beigins with the Pythagorean Theorem. It is referred to the Pythagorean Ideinity because we use the Pythagorean Theorem to get it.

As you can see in the picture above, I chose one of the "Magic 3" ordered pairs from the unit circle to show that it is true. The ordred pair used was the 60* angle one which is (1/2, rad3/2).

2.

The picture above shows that how to derive the identity with Secant and Tangent. First we had to divde everything by cos^2ø. Then that would cancel out the cos^2ø to make that be 1. Then we got sin^2ø/cos^2ø which we subtitute with tan^2ø. Lastly we have 1/cos^2ø and we substitute that with sec^2ø. Our final answer will look like tan^2ø + 1 = sec^2ø. 

The picture above shows that how to derive the idenity Cosecant and Cotangent. First we had to divdie everything by sin^2ø. Then that would cancel and make it be 1. After we got cos^2ø/sin^2ø which we substitute with cot^2ø. Lastly we have 1/sin^2ø which turns into csc^2ø. Our final answer will look like 1 + cot^2ø = csc^2ø.