BQ#2 – Unit T Concept Intro

How do trig graphs relate to the unit circle?
Trig graphs are largely related to the unit circle! This can be seen in the formation of the graphs and the curves that accompany specific trigonometric functions. Each trig function correlates to a quadrant on the unit circle. This creates zones of positive and negative values. These values are reflected throughout trig graphs as the curvature of the lines follows the specific guides. The "all students take calculus" technique still applies! If we were to unravel the unit circle, the positive and negative quadrants would still apply to the trig functions.

Period?- Why is the period for sine and cosine 2π, whereas the period for tangent and cotangent is π ?
The period for sine and cosine is 2π because that is how long it take to repeat the pattern! The pattern that applies to sine is "++--" while cosine refers to "+--+" In order to complete these patterns (or make a full rotation around the circle) you would need to use all four quadrants. As we know from our unit circle knowledge, a full revolution around the circle is 360 degrees OR 2π! That being said, it would take 2π to complete the necessary pattern. The period for tangent and cotangent is π because you only need half of the unit circle to create a period. This is because the tan and cot pattern can be created with only half of the unit circle; half of the unit circle is equal to π.

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?

In order to answer this question, we should think back to what we know about the unit circle. From previous lessons, we have learned that the unit circle has a radius of 1. We have learned that the trig function for sine is equal to "y/r" and cosine is equal to "x/r." If "r" - the radius - is 1, that means that the numerator would have to be a value that is less than 1. We also know that sine and cosine have limitations within the circle. The two must be smaller than both -1 and 1. If sine and cosine cannot pass the value of "1" in either direction on the unit circle, they won't be able to pass them on a regular graph (remember that a trig graph is just the unit circle unraveled). That being said, the two trig functions are restricted to the y values of 1 and -1. This is the equivalent to having amplitudes of 1 and -1.