Unit T Big Question #1

How do the trig graphs relate to the unit circle?

Part 1a:
Why is the period for sine and cosine 2pi, wheras the period for tangent and cotangent is pi? 

•  For sine and cosine graphs, it takes 4 parts in order for its "pattern" to repeat. For instance, the picture shows sine's pattern: ++--. That is one period. The period is marked at the point that the pattern is about to repeat. This basically means that for sine and cosine, it will take a whole rotation of the unit circle in order for the graph to repeat itself. (Hence, 4 parts and 4 quadrants). This is one period. Therefore, for sine and cosine, a period goes through 2pi units on the x-axis (also 360* and one revolution of the unit circle.)

• Tangent and cotangent are different. While their pattern goes + - + -. It only takes two parts for the graph to repeat itself: + - and then we go again with + -. We do not need the other two "parts" or we would be going into a SECOND period of the graph. Therefore, we only need half of the unit circle( pi and also 180*) to cover a cycle. One period for tangent and cotangent would cover only two "parts" and only pi units on the x-axis.

  Part 1b: How does the fact that sine and cosine have amplitudes of 1 relate to what we know about the Unit Circle?
  *First of all, we must remember that amplitude is HALF the distance between the highest and lowest points on the graph. *
  

  If we look closely, the Unit Circle resembles the graph greatly. That is because the Unit Circle in "line" form would be that graph and it is like we are stretching it out. If we notice, sine has two amplitudes here. One at pi/2 and another at 3pi/2. We notice that on the Unit Circle at those exact values the sine is 1. That is where we get our amplitude from.  
  
  
  
  As you can see, cosine has three amplitudes. At 0, at pi, and at 2pi. If we look at the Unit Circle, cosine begins at 0, goes halfway to pi and then ends at 2pi. If we notice, the cosine of those values is 1 and -1. Which correspond with the graph. The Unit Circle is a great reference to make sure your graphs are on point as the graphs are simply just the Unit Circle laid out flat.
  
  

  
  
  

  Since cotangent, tangent, cosecant, and secant all have asymptotes, they do not have amplitudes like sine and cosine. As you can see, where sine and cosine equal zero, there will not be amplitudes but instead....ASYMPTOTES.