How do the trig graphs relate to the unit circle?
Trig graphs relate to the unit circle in how the unit circle can be utilized to find where a certain trig function is positive or negative on a graph. For example, sine is positive in the first and second quadrants - between 0° and 180°, or between 0 and π. If we utilize radians on our x-axis, and have some fraction of π as our markings, then we may isolate the area of the graph between 0 and π as the area in which sine would be positive. Finding these positive/negative areas works with every trigonometric function, though you would have to account for asymptotes as well for denominators that could possibly be equal to 0.
Period?
Why is the period for sine and cosine 2π, whereas the period for tangent and cotangent is π?
The periods for sine and cosine are 2π because it takes a full rotation of the unit circle, which is 2π, for sine and cosine to fulfill the requirements to be periods. And, a period is a portion of any trig graph that is repeated over and over again without change. To illustrate this example, the values of sine, going through the unit circle's four different quadrants, should be +, +, -, -. Having anything less than those four repeating themselves would create inconsistency, so there you have it. On the other hand, the periods for tangent and cotangent are π because it takes half the rotation of the unit circle. π, for tangent and cotangent to continually repeat themselves consistently. The values of tangent/cotangent, going through the unit circle's four different quadrants, should be +, -, +, -. And having just two repeat themselves continually should be enough to constitute an acceptable pattern.
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?
Sine and cosine have amplitudes, while the other trig functions don't have amplitudes, because their ranges are restricted by how both of their trigonometric ratios have 'r' as the sole value in their denominators. And, on the unit circle, 'r' is equal to 1, being the length of the radius. The other trig functions don't have amplitudes because restrictions such as 'r' aren't in their ratios.
