1. How do the trig graphs relate to the Unit Circle?
a. period?- why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?
b. How does the fact that sine and cosine have amplitudes of one (and other trig functions don't have an amplitude) relate to what we know about the Unit Circle?
Answers:
1. The trig graphs relate to the unit circle by them following the same repetitive patter when going in a circle yet not it is shown as a line on the x axis. For example, for the trig graphs the quadrants used in the unit circle are now drawn on a line rather than a circle where the sign changes for each different trig function still remains the same.
In this picture you can see how the renewed unit circle looks and how familiar it looks to the actual circle by containing the three major points (90, 180, 270, and 360) all on this axis which does not lose its significance when figuring out the signs of each trig function. The signs remain the same as when we figured them out using ASTC, if we even wanted to we could still write ASTC on it and still be able to use it.
a. The period relates to the unit circle because one revolution around the unit circle is basically one cycle in which it takes for the pattern of signs and everything to repeat itself. So just like a period on these graphs it shows or lets us know how long it takes the trig function to go through one cycle. The period for sine and cosine are 2pi because thats how long it takes to go through one time of their cycle for it to then repeat itself again. They cover 2pi units on the x axis before repeating themselves all over. This picture shows just that, the pattern from 2pi repeating over again once going through a cycle.
As for cotangent and tangent the period is pi because again that is how long of a cycle it takes for one cycle, before the pattern of these two repeat itself. They have to cover pi units in order to start repeating there cycle once again.
b. The trig graphs relate to the amplitudes by them dealing with the numbers dealing with the coefficient that is connected with the trig function. The fact that sine and cosine have amplitudes relate to what we know about the unit circle because on the unit circle we are familiar that at each reference angle 0, 90, 180 and 270, they have points connected to them and when looking at the points at 0 is (0,1) and at 180 its (0,-1). These two points on the graph simply show how sine is 1 because it starts at 1 and ends at 1 for one cycle on the Unit Circle. As for cosine this is evident through the points at 90 and 270 where 90 has the points (1,0) and 270 has the points (-1,0), through this knowledge from the unit circle it is easy to see how both cosine and sine have amplitudes of one because it is dealing with the ordered pairs and the cycle needed for one of the trig functions. The other trig functions can also be explained to have no amplitude because if you think about it for tangent you would either be getting undefined or 0 as the amplitude, same goes for cotangent.
Citations
Picture 1:
http://mathbits.com/MathBits/StudentResources/GraphPaper/trigsmaller.jpg
Picture 2:
http://img.sparknotes.com/figures/A/ad79275cb59e569b790cb945a4ffc553/quadrantgraph.gif
Picture 3:
http://www.s-cool.co.uk/a-level/assets/learn_its/alevel/maths/trigonometry/graphs-of-trigonometric-functions/2007-10-09_133331.gif
Picture 4:
http://www.calculatorsoup.com/images/trig_plots/graph_cot_pi.gif
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