Unit T Big Question #1

How does the trig graphs relate to the Unit Circle?
A. Period? Why is the period of Cosine 2 pi, whereas the period for tangent and cotangent is pi?

1.      What is this concept about?

This concept is about is understanding why there is a difference among the periods of sine and cosine and tangent.  These pictures will show the relationship the period has with the unit circle.  By doing this we learn the pattern cosine, sine and tangent have on the Unit Circle which helps us with answering our question.

What the viewer needs to pay close attention to in order to understand the concept?

The viewer needs to pay close attention to the relationship among the unit circle and the graph. By this I mean that they are similar because the graph takes the same units on the quadrant angles (0, π/2, π, 3π/2, 2π) and lays them horizontally on the x- axis.  A KEY FACTOR is to know the pattern of each ratio, this helps tremendously with understanding the answer to our question.

PATTERN:

 Sin++--

This picture depicts how the unit circle lies on the X axis with it s units. It is also showing how the patterns go for the whole period from 0 (0 degrees) to 2 pi ( 360 degrees), 1 whole revolution. 

Cos +--+

For the pattern seen above the highlighted area shows the repetition the tangent pattern within the revolution, therefore since it goes half a revolution the period is pi because that is where it starts its repetition. 

Tan    +-+-

1. B. 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 on the Unit Circle? 

Amplitude: Is the" half distance between the highest and lowest point".

         What is this concept about?

This concept is about explaining why sine and cosine have amplitude of 1 whereas the other trig functions don’t. This is going to involve the Unit Circle and its ordered pairs, it contains on the quadrant angles.

What the viewer needs to pay close attention to in order to understand the concept?

Pay close attention to the explanation on how the unit circle and the graph are related. Knowing where the ordered pairs will be located on the graph is very important. As for the other trig ratios they are derived from the cosine and sine ratio identities and reciprocal identities in order to get  the asymptotes. In this case all the trig ratios besides cosine and sine will have asymptotes. 

Tangent, Cotangent, Secant and Cosecant don't have an amplitude because they don't have a  highest or a lowest point on there graph, therefore no amplitude can be found.  What they do have is asymptotes, which are explained in Unit T Big Question #4.(Pictures shown below show cosecant and secant graphs)

   

This image shows the ordered pairs of the quadrant angles  on the unit circle. The highlighted numbers are the ones we will use for labeling the graph.

Here the graph is showing the amplitude of the cosine which is one. In the previous picture
shown it highlighted the numbers we were going to use. The graph has points on 1(0), 0 (pi/2), 1(pi), 0(3pi/2),1(2pi). But we know the amplitude is one because at the ordered pairs  (0,1) ,(pi,-1),(2pi,1) have the amplitude of one.   On the bottom it shows where the line will cross to be at 1, the amplitude. 

Similar to the cosine graph the amplitude of sine is  1. The amplitude is located wherever sine is 1 on the ordered pairs. The sine is one at the 90 degree, which is pi/2 on the graph and at  270 degrees which is 3pi/2 on the graph. With knowing this the graph will have a point on (pi/2,1) which gives it the amplitude of one. An the other point is (3pi/2,-1) which has the amplitude of one as well. The bottom shows where the line will cross the one.


Cite:
Unit T SSS Packet
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