BQ #2: Unit T: Concept Intro: How do trig graphs relate to the Unit Circle?

How do the trig graphs relate to the unit circle?
The Unit circle relates to the trig functions because it helps describe where the positive and negative sections of the graph come from, so that we can derive the periods of each trig function in order to graph it. When you look at the graph below, notice how the sine (Blue) is positive, which is basically covering the 1st and 2nd quadrant because sine is positive there. Then, it continues to go down to reflect that sine is negative in the 3rd and the 4th quadrant. As for cosine (Green), the positive uphill tells us that cosine is positive in the 1st quadrant and then goes down because cosine is negative in the 2nd quadrant. The downhill also shows that cosine is negative in the 3rd quadrant since it is still negative. It finally goes back up because cosine is positive in the 4th quadrant. The tangent graph (Orange) also goes through the similar situation, but according to the ASTC pattern as well. So first the graph starts in the 1st quadrant because all trig functions are positive there and it becomes negative since tangent is negative in the 2nd quadrant. It finally goes back up to a positive position because tangent is positive in the 3rd quadrant and then goes back to negative in the 4th quadrant.

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

Let's start by defining what a period is in order to understand how it relates to the graphs.
Period: The period for sine/cosecant and cosine/secant is 2 because it will take 4 quadrants (based on ASTC) to repeat the pattern (or one time to go through the cycle)

Sine is + + - -

http://geogebratube.com/student/m25914

Cosine is + - - +

http://geogebratube.com/student/m25914

Sine and Cosine may have an amplitude of 2, the period for Tangent and Cotangent is simply  because tangent has a pattern on +-+- (based on ASTC).  In other words, it has a positive and then a negative within the first two quadrants.

 Tangent + - + -

http://geogebratube.com/student/m25914

 Remember, a period is only how much it take to go through the cycle even though it is never-ending; in fact, periods can continue on forever, but we only want to look at one period in order to understand how it relates to the trig functions.


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?

Amplitudes are half the distance between the highest and lowest points on the graph. All we need to look at is the value of "a" to find it (Kirch). So why is it that sine and cosine have an amplitude, but not tangent? Well, going back to the unit circle, sine and cosine both have "r", which equal 1 in their trig ratios. Sine and cosine have a restriction because they cannot have a value greater than 1 or less than -1. In fact, this would result in a "no solution". However, tangent/cotangent do not have 1 and are not restricted to 1 and can have other values that do not have to be in between -1 and 1.

https://share.ehs.uen.org/taxonomy/term/300

References:
https://share.ehs.uen.org/taxonomy/term/300
http://geogebratube.com/student/m25914
Unit T SSS Packet