 Trig Graphs and their Relation to the Unit Circle
 Trig graphs relate to the unit circle because their signs depend on what quadrant they are in when looking at a unit circle. For example a sine function will be positive in quadrants I and II while being negative in quadrants III and IV.
2. Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
 The period is 2pi for sine and cosine because of their pattern on the unit circle repeats after every four marks on a graph. If we measure our xaxis by pi/2 each mark, it would take a full revolution around the unit circle because it has 4 of these as quadrant angles each represent a pi/2 and for sine and cosine, the sign pattern is +,+,,. Which means that it won't repeat after it has passed four marks. For tangent and cotangent ,however, its pattern is +,,+, and you can clearly see that it repeats just by going to two marks. Two marks on our xaxis would equal to just one pi.
Look at these sine and cosine graphs for example and notice their quadrant signs and how they go all the way to pi.

(http://www.regentsprep.org/Regents/math/algtrig/ATT5/unitcirclecowine.gif) 

(http://www.regentsprep.org/Regents/math/algtrig/ATT5/sine77.gif) 
Now look at these tangent and cotangent graphs and notice that they immediately start repeating just after two marks, or just pi on the xaxis.

(http://www.analyzemath.com/trigonometry/graph_cotangent.gif) 

(http://www.analyzemath.com/trigonometry/graph_tangent.gif) 
3.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?
 Sine and Cosine have amplitudes of one because they have restrictions on the range of their graphs. When we look at tangent and cotangent they go up to infinity and down to negative infinity because they don't have these restrictions. But sine and cosine have the restriction of 1 because on the Unit Circle simple because anything greater than 1 for sine and cosine is no solution. Here's an image showing the difference in range between a sine graph and a cosecant graph.

(http://learn.roguecc.edu/mathematics/chutt/Math112/Intro/OtherTrigonometricFunctionsDay1_files/image004.png)
References

http://learn.roguecc.edu/mathematics/chutt/Math112/Intro/OtherTrigonometricFunctionsDay1_files/image004.png
http://www.analyzemath.com/trigonometry/graph_tangent.gif
http://www.analyzemath.com/trigonometry/graph_cotangent.gif
http://www.regentsprep.org/Regents/math/algtrig/ATT5/sine77.gif
http://www.regentsprep.org/Regents/math/algtrig/ATT5/unitcirclecowine.gif