I/D#3: Unit Q Concept 1: Pythagorean Identities

INQUIRY ACTIVITY SUMMARY

1. The Pythagorean Theorem is an identity because it is a proven fact that is always true. The Pythagoreom Theorem is a^2+b^2=c^2, but in the Unit Circle, these letters are mentioned as x, y and r, making a slight shift to the original theorem. With these letters, the Pythagorean Theorem is now x^2+y^2=r^2. In order to make the Pythagorean Theorem equal to one, we must divide both sides of this equation by r^2, as seen in the picture below.

 

Then, going back to the Unit Circle,  the ratio for cosine was x/r and the ratio for sine was y/r. These ratios can then be plugged into (x/r)^2+(y/r)^2=1 in order to get sin^2x+cos^2x=1. This is referred to as a Pythagorean Identity because it is just simply the Pythagorean Theorem rearranged in a different way. To show that this is identity is true, we can use one of the "Magic 3" ordered pairs from the Unit Circle. If we have a 45 degree angle, we know that the ordered pair is (radical2/2, radical2/2). When this is plugged into the equation, it will be radical2/2^2+radical2/2^=1. This is true because when radical2/2 is squared, it results to being 1/2 and then when it is added to the other 1/2, it is one. Therefore, the identity is true.

2.To derive the identity with Secant and Tangent, cos^2x must be divided by both sides in the equation of sin^2x+cos^2x=1. You will get:

 

In the unit circle, sine has a ratio of y/r and cosine has a ratio of x/r. When the ratios are divided,with sine being the numerator and cosine being the denominator, you get y/x, which is the ratio for tangent. The two cosines will then cancel and result in being 1. 1/(x/r)^2 (the ratio of cosine) is equal to sec^2. This is how you get tan^2x+1=sec^2x.

To derive the identity with Cosecant and Cotangent, we must now divide sin^2x+cos^2x=1 by sin^2x.

 

The sines will cancel, leaving us with 1. The ratio of x/y equals cotangent (refer to picture for further explanation). 1/sin^2x equals csc^2x because of reciprocal identities. This ends up being 1+cot^2x=csc^2x.   

INQUIRY ACTIVITY REFLECTION

 

1. "The connections that I see between units N, O, P, and Q so far are..." that they all somehow connect to the Unit Circle and also that they somehow involve triangles as well.

 

2. "If I had to describe trigonometry in three words, they would be..." complicated, intricate, and time-consuming.