BQ #2: Unit T Concept Intro

How do the trig graphs relate to the Unit Circle?

The trig graphs belive it or not relate to the unit circle in a big way. One period of a trig graph is one full revolution of the unit circle, no matter what. The trig graph relating to the Unit Circle in the way that it  indicates when the trig graph is negative (down sloping) or positive (up sloping). For example; when looking at sine in terms of the unit circle it's pattern is positive, positive, negative, negative. This means that everytime it goes through one cycle while covering 2pi units on the x-axis. That is as to why the sine trig graph looks the way it does. Same situation applies to cosine, cosecant, and secant. Cosine's pattern being: positive, negative, negative, and positive. Although the only exception is tangent/cotangent who instead goes through one cycle covering pi units on the x-axis. Tangents pattern being: positive, negative, positive, negative.

Period? - Why is the period for sine and cosine 2pi, whereas the period for tangent and cotangent is pi?

-The period for a sine and cosine covers 2pi units on the x axis, because as seen in the image above in the sine pattern that goes positive, positive negative, negative. Which means it goes up slopping, up sloping, down slope, down slope and because of that it doesn't cross the x axis till 2pi. Same going for cosine.

While for tangent/cotangent it covers pi units on the axis, because again as seen in the image above the tan pattern is positive, negative, positive, negative. Meaning that it goes up slope, down slope, up slope, and down slope which gives us have a more squished graph. Unlike the sine and cosine graph which appears to be a bit more apart.

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 because when referring to the ratios of each sine has a ratio of y/r and cosine has ratio of x/r. R being the radius of the unit circle, which is 1. That is as to why sine and cosine can be no larger than one or negative one, as said so in past units. But when dealing with other trig functions and the ratios they have the same doesn't apply to them. For instance tangent has a ratio of y/x meaning that we no longer are dividing by one, instead we dividing by different numbers, thus getting different values. The same thing applying to cotangent, who has a ratio of x/y. This also applies to csc with ratio of r/y and sec with ratio of r/x. Which can be divisible by fractions, and proceeding by multiplying with the reciprical of the denominator.