BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the unit circle?
Trig graphs, believe it or not, are essentially the unit circle unwrapped (at least in one period). BAM. Let's think about what we already know about the unit circle. We are going to use sine as an example, first. In the case of sine, the 1st and 2nd quadrants are positive whereas the 3rd and 4th quadrants are negative. Now, the 1st quadrant goes from 0 radians to pi/2. The 2nd quadrant goes from pi/2 and pi. So from this information alone, we can deduce that from 0 to pi, our values are going to be POSITIVE. The 3rd quadrant goes from pi to 3pi/2; the 4th goes from 3pi/2 to 2pi. Because sine is negative in these quadrants on the unit circle, in the trig graphs, from pi to 2pi, our values are going to be negative. Thus, the whole entire graph will go up and then down, because it is corresponding with the values we know from the unit circle.

 

As you see, the same can be seen for other trig functions. For cosine, we have the pattern being positive, negative, negative, postive. Therefore, on the graph, you will see the period beginning with positive values, but dipping at pi/2 and becoming negative. At pi, it remains negative, but when you hit the mark of 3pi/2, our values will be positive, as it should.

 

Tangent and cotangent are similar however there is a special situation with them (which I will go into further detail in the next section). Anyway, our graph for tangent/cotangent will begin with positive values, then negative values, then positive, and negative. However, if you notice, we have a pattern already seen in the first two quadrants, that is then repeated in the next two quadrants--which leads us into the next question!

 

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 is 2pi because the pattern finishes at 2pi and does not repeat until we begin another "revolution" around the unit circle. The pattern is +,+,-,- for sine. The pattern is +, -, -, +. You see, you need all quadrants in order to explain the FULL pattern. You need to go around the unit circle in ONE FULL revolution. So, your period will go to 2pi.

The period for tangent and cotangent is only pi. Why? Look at the picture above. Do you notice that the pattern is +, -, +, -? The pattern repeats itself twice in one revolution of the unit circle. Why do we want to repeat that? So, we only really need the first quadrant in order to fully explain the pattern. That is why tangent and cotangent periods only go to pi.

 

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?

So why is it that we have amplitudes of one for sine and cosine? Think about it. Sine has the ratio of y/r. Cosine has the ratio of x/r. R can only equal 1, since the radius of a unit circle has to be 1. So, the values x and y can only go up to 1 themselves. So if you can only divide 1 by 1 to get the largest number... your largest value can only be 1/ -1. That is why sine and cosine cannot equal something greater than 1 or less than -1. However, think about the other trig functions and their ratios. Tangent is y/x. You are not restricted to values between -1 and 1 because you are dividing by different numbers that vary from being just 1 and -1. Same is for cotangent but backwards. Also, for trig functions like cosecant and secant, you have r/y and r/x. You can divide your "r" (1) by a small number like 1/2 and end up with 2. That is why those other funcitons do not have amplitudes.