Trig graphs relate to the Unit Circle because if we took the Unit Circle and unwrapped it, it will become a straight line and the four quadrants will become hash marks.
A.)So, for sine and cosine the period is 2 pi because sine's pattern is ++-- and cosine's pattern is +--+(based on All Students Take Calculus) and looking at these patterns it will take the whole rotation of the Unit Circle or all of the four quadrants for the pattern to begin to repeat itself. If we look at the Unit Circle half way at 180 degrees it is pi and at 360 degrees (which is all the way around) it will be 2pi and since it takes sine's and cosine's pattern to begin repeating itself after four quadrants, which is at 360 degrees and 360 degrees is 2pi. One time through their cycle is called a period and therefore sine and cosine have periods of 2pi. On the other hand, tangent and cotangent have periods of pi(180 degrees) because tan's and cot's pattern is +-+- and as you can see the pattern is repeated twice. So, after two quadrants(180 degrees) the period starts to repeat itself therefore it is pi, instead of 2pi.
B.)Knowing the fact that sin and cos have to be in between 1 and -1 and that the Unit Circle can extend (0,1 at 90 degrees), (0,-1 at 270 degrees), (1,0 at 0 and 360 degrees), and (-1,0 at 180 degrees) we can state that sine and cosine have amplitudes of one. Also, sin and cos both have r=1 as a denominator.