Being near the end of the second semester in this mighty Math Analysis class, we are all familiar with the seemingly endless circle of power that is the Unit Circle. This same Unit Circle can be transformed into the trigonometric graphs we have all come to love greatly by now. We achieve this by simply unwrapping the almighty Unit Circle and placing it in the shape of a simple graph.
-Period? - Why is the period for the sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?
The reason for this is simple. In the sine and cosine graphs, you have your "mountain" that takes up basically quadrants I and II, while the "valley" takes up quadrants III and IV. Since the pattern is technically +|+|-|-|+|+|-|- and repeat for sine, it takes you two negatives to get the back to the double + pattern, thus the 2 pi. In the case of cosine, the pattern is
+|-|-|+|+|-|-|+|. As you can see, it still takes 2 negatives before you can get to 2 positives, thus the 2 pi. In the case of tangent and cotangent, the pattern is +|-|+|-. This means that you only have to get through 1 negative before you get to the positive.
-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 he Unit Circle? Basically, both Sine and Cosine are (respectively) y/r and x/r, which gives is basically y/1 and x/1, thus setting the the restriction of 1 and -1. On the other hand, Tangent and Cotangent are x/y and y/x. This means that we do not have any restrictions because we divide by numbers other than 1 and -1.
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