How do trig graphs relate to the unit circle?
Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The period for sin and cos is 2pi because of the patterns they follow on the unit cirlce. Sin follows a ++-- pattern and cos follows a -++- pattern. It takes one full unit circle in order for the pattern to repeat for sin and cos and one full unit circle has a length of 2pi. Since a period is how long the graph takes to reach the point it started it will take a length of 2pi because that's how long it takes the patterns for sin and cos to repeat themselves. The period for tan and cot are both just pi units long. This is because the unit circle pattern for tan and cot is +-+-. Because of that pattern it takes only half the distance of the unit cirlce to repeat itself. Half the distance of the unit circle would be pi, making the period for tan and cot just pi.
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?
For trig functions only only sin and cos have amplitudes because of they behave on the unit circle. The limits of a cos on the unit circle are found at 1,0 and -1,0. The +1 and -1 look very familiar. That's because the amplitude of a cos graph is alos equal to one. Since the limits of cos are at those coordinates that has to also be the limits of a cos graph. Since 1,0 is the start and end of a the unit circle that's why cos starts and ends at 1. The same goes for sin only sins limits are found at 0,1 and 0,-1. Since the limits are again at 1 and -1 on the unit circle the same must be the same for the trig graphs.