BQ #2: Unit T Concept Intro

How do trig graphs relate to the Unit Circle?
Ultimately, we can say that a trig graph, in its first period, is just a different presentational form of the Unit Circle that has been unwrapped on a graph. In referring to the original form of the Unit Circle, we know that each trig function has two quadrants where they are positive and two where they are negative. For example, sin is positive in both the first and second quadrants but negative in the third and fourth. By knowing that they are positive in the first and second, we know that the distance from the beginning of Quad 1 and Quad 2 is 0 to pi; the distance from Quad3 to Quad 4 is 3pi/2 to 2pi where we know sine is negative. What a trig graph generally represents is the sign change for all trig function values by stating if the values are positive (above the x-axis) and if they are negative (below the x-axis). This forms wave like motions on the graph.

 

The trig graphs will go on forever repeating the same period for each trig function.

 



Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
Sine and Cosine both have a period that end at 2pii because that is when the pattern ends. A period is the end to a pattern on the graph. Because Tangent has a pattern of positive-negative in the first two quadrants and repeats for the third and fourth, we know that the period for tangent and cotangent are cut down to half which is just pi.

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?
So why is it that sine and cosine have an amplitude of one? Maybe the Unit Circle can answer that. When we look at the trig functions sin = y/r and cos = x/r; because it is the unit circle, we automatically know that r is the value of one. So it is because we may only divide by one that we must assume that the move must be a value of one or negative one. This also explains why the trig of sin and cosine may not equal any value that does not lie between one and negative one. However, these do not apply to any other function. Why not, you ask? Well let's look at tangent; the ration for tangent is y/x. There are no restriction values for either y or x which means that there is a possibility for the number to exceed the distance of one in either direction. the same applies for the other trigs.