BQ #2: Unit T Concept Intro

How do the trig graphs relate to the Unit Circle?

1. 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 2pi because, going around the Unit Circle, their patterns take the whole circle to repeat again. For example, sine's pattern is positive positive negative negative. Since there is no clear pattern, we must use the whole Unit Circle to repeat it again and thus, the period is 2pi. The same can be applied to cosine's pattern around the Unit Circle, which is positive negative negative positive. Similarly, it has to be repeated again for a pattern to emerge, so the period for cosine is 2pi (the distance around the Unit Circle). 

Tangent and cotangent, on the other hand, have a clear repeating pattern without having to go all the way around the Unit Circle: positive negative positive negative. We see that "positive, negative" is the repeating pattern and that it repeats itself halfway around the Unit Circle. Then, it starts over again. As a result, the period of tangent and cotangent is only pi, which is half the distance around the Unit Circle.

2. 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 having amplitudes of one relates to the Unit Circle in an important way. First of all, to understand the amplitudes of sine and cosine, we would have to know that the range of the Unit Circle values. They are between 1 and negative 1. And so, with sine having a ratio of y/r--or simply 'y'--the graph's amplitude is limited to one. Thus, the amplitude will remain at one. Cosine, similarly, has a ratio of x/r--or simply 'x', since r is 1--and its graph will have an amplitude of 1.