How do Trig Graphs relate to the Unit Circle?
a) Period-Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?
According to the unit circle, sin is positive in quadrants I and II whereas quadrants III and IV are negative making the pattern +, +, -, -. Cos is positive in quadrants I and IV whereas quadrants II and III are negative making the patter +, -, -, +. It takes a whole unit circle to complete a pattern until it repeats itself. When you look at a unit circle, the circumference is 2pi. If you cut it at one point and laid it flat, it would still be a length of 2pi. That is basically a sine and cosine graph's period. You can also see the patterns represented on the graph.
b) 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?
This is because in the unit circle they are restricted to (0,1), (-1,0), (1,0), (0,-1) or restricted from -1 to 1 which is the radius of the Unit Circle. This creates walls for the sin/cos graphs and thus they have amplitudes of one.