How do the trig graphs relate to the Unit Circle?
A trig graph can be seen as an 'unwrapped' Unit Circle. The quadrants of a Unit Circle correspond with the sections on a trig graph. Whether or not the section of the graph is positive or negative -in other words, above or below the x axis- depends on the trig function graphed. For example, since sine is positive in Quadrants I and II and negative in Quadrants III and IV, the pattern of the sections of the trig graph for sine is + + - -, or above the x axis for two sections and below for the next two sections.
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 it takes a length of 2pi along the x axis of the graph to complete the pattern of positives and negatives for sine and cosine. This length is also the length of the circumference of the Unit Circle (how coincidentally convenient!) The period for tangent and cotangent, on the other hand, is pi because it only takes a lengths of pi along the x axis of the graph to complete the pattern of positives and negatives for tangent and cotangent. Since tangent (and cotangent) is positive in Quadrant I, negative in Quadrant II, positive in Quadrant III, and negative in Quadrant IV, it only takes half of the circle to repeat the 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 because their waves are restricted. On the Unit Circle, sine and cosine cannot be bigger than 1, but neither can they be smaller than -1. The other trig functions don't have these restrictions in the Unit Circle, so they don't have restrictions in their graph either in the form of amplitudes.
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