ID#3: Unit Q Concept 1: Pythagorean Identities

The pythagorean Theorem is an identity because it is a proven fact or formula that is always true. The derivation actually comes from the unit circle. In quadrant 1 of the unit circle we get a triangle with all positive values and the x-axis is x, the hypotenuse is r, which equals 1, and the other side is y. The first thing we do is set up our equation, which is x^2+y^2=r^2. Next we divide both sides by r and the r will cancel, leaving us with 1, and our equation should be (x/r)^2+(y/r)^2=1. With this information we can tell that x/r is cosine, and y/r is sine, so now our equation should be cos^2x+sin^2x=1.

One of the next two identities is derived from the first identity. The first thing we do is divide all sides by cos^2x and the equation should be (cos^2x/cos^2)+(sin^2x/cos^2x)=(1/cos^2x). Cos^2x/cos^2x cancels and becomes 1, thus leaving us with 1+(sin^2/cos^2x)=(1/cos^2x). From our identities we know that sin^2x/cos^2x is the same as tanx and that 1/cos is the same as secx and we end the final equation of 1+tan^2x=sec^2x.

For the last identity we must divide everything by sin^2x and we should end up with (sin^2x/sin^2)+(cos^2x/sin^2x)=(1/sin^2x). (sin^2x/sin^2x) cancels out and leaves us with 1, so our equation should be 1+(cos^2x+sin^2x)=(1/sin^2x). From our identities we know that cosx/sinx is the same as cotx and that 1/sinx is the same as cscx. Our final identity should end up being 1+cot^2x=csc^2x.

The connections I see between units N, O, P, and Q are that most of the information comes from the unit circle, such as the trig functions. 

If I had to describe trigonometry in THREE words, they would be tricky, cool, and cornfusing.