BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the unit circle?

Trig graphs relate to the unit circle by the quadants and the positive and negative values. The unit circle is bascially just unraveled onto a line. Since we know that each quadrant as a corresponding value according to it's trig function, the direction of the graph will follow. For example, sine in a Unit Circle is only positive in the first and second quadrant, then in the third and fourth quadrant, it is negative. When it is drawn out on a graph, we know the position of each parts of the line will either be on the positive side or negative. This means that sine will be on the positive the first two parts, then it will be on the negative side. The image below indicates the positive and negative values in each quadrant for each trig function. Following the color coordination and +/- values, the line drawn below shows where each part of the quadrant exists.

(https://docs.google.com/file/d/0B4NSkh2FgPbXQ2p6dmxpdUVLa1k/edit)
*Drawing and labeling done by me*

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

The period of sine and cosine is 2pi because it would take a whole circle to complete the line. We see that both either stay on the positive side or negative side for a good two quadrants. Then, it ends up at where the starting point is. Since the cycle goes through all four quadrants, we know that in a Unit Circle, a 360' would be 2pi. That is why the period for sine and cosine is 2pi. 

The peiod for tangent and cotangent differs. It is just pi. In the image above, it shows how tangent would be graphed on the line. It goes from a positive value, then dips all the way down (representing by an asymptote) towards negative and goes back up the the positive, then back down. From this, we only need to go up to the second quadrant because the cycle repeats itself from there on. In the Unit Circle, we know that the second quadrant goes up to 180', which in radians is just pi. This is why tangent and cotangent's period is pi.

Amplitude? – How does the fact that sine and cosine have amplitudes of one (and the other trig functions don’t have amplitudes) relate to what we know about the Unit Circle?

We know that sine has a ratio of y/r and cosine has a ratio of x/r. The "r", or radius, will always equal to 1. Another thing to remember is that sine and cosine can't be greater than 1 or less than -1 or it would be "undefined". So the only way to divide a number from 1 and get either 1 or -1, is 1. This is why sine and cosine have an amplitude of 1.

Unlike sine and cosine, tangent, cotangent, cosecant, and secant don't have an amplitude. These all have ratios of either x/y, y/x, or r's over x and y. There is no limit to them being 1 since the radius is not on the bottom. In tangent, y/x, the value of the varibles can vary and it is not restricted; it doesn't matter whether it equals to  -1/1. That is why these trig functions do not have amplitudes.