BQ#2: Unit T Intro

How do the trig graphs relate to the unit circle?

A trig graph is actually a unit circle. Well, not the whole thing since the graph goes on forever, but from 0 to 2π, which is essentially the length around a whole unit circle.
To test this, we can use our trig functions to help. For example, lets use sine. For sine, it is positive in the first two quadrants and negative in the last two. Now, how is this related at all? Well, when we uncoil the unit circle and make a trig graph for sign, essentially what is quadrant one and quadrant two will be above the x-axis and will be positive. What would be the third and fourth quadrant would be below the x-axis since it is negative.
Make sense yet? Somewhat? Let's try another one! Let's use cosine.Cosine is positive in quadrant one and quadrant four and negative in quadrant 2 and 3. Now, let us once again imagine the unit circle unwinding and placing it on a graph What would have been quadrant one and quadrant four would be above the x-axis because it is positive for cosine in those quadrant. Now, the second and third quadrant will be below the x-axis since cosine is negative in those quadrants.
Making more sense? Need just one more example? Let's use tangent! Tangent, is positive in the first and third graph and negative in the second and fourth graph. Now, for the last time, uncoil that unit circle and think, how would our graph look like? ... Quadrant one and three would be above the x-axis because it's positive and quadrant 2 and 4 would be below the x-axis since it is negative.

©Kelsea DC

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

Lets refer back to what we reviewed above to help us. Also, note that a period is when a part of the graph reaches both above the x-axis and below the x-axis. As we know, sine, cosine, and tangent are positive in certain area. Sin: ++--; Cos: +--+; Tan:+-+-. These are the patterns for these. Now a period is how much it takes for something to repeat. When we look at sine, how long does it take to obtain a period of ++--? Well, it takes 2π and also because there are no repeating factors in the format and that how long it would take for it to repeat again. . For cosine, what kind of period would it take for +--+? Well, 2π again since there are no repeating factors and that's how long it takes in order for it to repeat again.  Now, tangent is +-+-. What's it's period? 2π? WRONG! It's π! Why? because in one period, the graph reaches both above the x-axis and below the x-axis. Need a visual? NO WORRY! Look Below!

©Kelsea DC
Green=Quadrant 1 Orange=Quadrant 2 Blue =Quadrant 4 Yellow=Quadrant 4

Amplitude?-How does the fact that sine and cosine have amplitudes of one (and other trig functions don't have amplitudes) relate to what we know about the unit circle?

Sine and Cosine can only have amplitudes of 1/-1 because on the unit circle, they are restricted to it. Their denominator in their equation is 1. If we try to use a number larger or smaller than it and plug it into the calculator, it becomes an error. However, tangent has an equation of y/x so it isn't restricted because y and x aren't really a specified number like r is. Same applies to cotangent, only the equation is flipped. 

Reference:

Mrs. Kirch's SSS Packet 

Hand Drawn Photos By Moi