So, you may ask, how do trig graphs relate to the unit circle? Well, it uses the positive and negative signs of the quadrants of the trig function! The trig graphs are basically the unit circle unwrapped or uncoiled to a line. The points on the graph are relevant to the points in the unit circle (radians). FOR EXAMPLE, in the unit circle, sine is positive in the first two quadrants, and negative in the third and fourth concepts (we know this from our previous units). Thus, on a trig graph, the graph will show a pattern of going uphill, uphill, downhill, downhill. This represents the quadrants. It is positive in the first two quadrants and negative in the other two. Lets use another example. For cosine, in the unit circle it is positive in the first quadrant, negative in the second, negative in the third, and positive in the fourth. So on the trig graph, the period is positive and then goes negative, and eventually goes back positive according to the marks. For tangent and cotangent, it will have positive values first, then negative, then positive, then negative. The trig graph has a shorter period and this is because in the first two quadrants, it goes positive then negative, and then it is basically repeated in the other two quadrants.
Beautiful visuals of the trig graphs and the patterns that I talked about in my earlier paragraph.
Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
The period for sine and cosine is 2pi because basically when you look at the unit circle, sine is positive, positive, negative, negative on the quadrants. In cosine, it is positive, negative. negative, positive. The period is 2pi and it takes all the quadrants (one whole revolution) to finish the pattern. But, for tangent and cotangent the pattern is positive negative. If it is positive and negative in the first two quadrants, and it repeats after that, then basically it only takes the period of pi in a revolution, in which it repeats itself twice.
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 of one. Why? Well think about what sine and cosine consists of. Sine is y/r and cosine is x/r. R is the hypotenuse, which is 1. Sine and cosine cannot be greater than one or less than one. On the unit circle, y and x are either 1 or -1. When you divide using the given ratios, it still equals to 1 or -1. So that clarifies that the amplitude is always one or negative one because of this rule. But, for other trig functions it is different. There are boundaries to what values it must equal to because x and y can vary in different numbers. For example, cosecant, it is r/y. Your y CAN be 1 or -1, OR it can be any other number like 2 or 3. Same goes to the other trig functions. So there are no amplitudes.