Unit T Big Question #2

How do the graphs of sine and cosine relate to each of the others? Emphasize the asymptotes in your response 
a. tangent
b. cotangent
c. secant
d. cosecant

What is this concept about?

This concept explains how sine and cosine are related to making the graph of the other trig functions. Ratio and reciprocal identities are used in order to understand this concept.

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

We need to pay close attention to the pattern of the signs in the graph. There is a trick that involves the signs to know whether the drawings on the graphs are correct for cotangent and tangent. For sin and cos the graph will lie in the corresponding ratios place. For example sin will stay with csc and cos will stay with sec.

Sine and Cosine are used to draw the other trig ratio graphs because they use the ratio identities and reciprocal identities to know the asymptotes to draw there graph. 

Tangent/Cotangent

The signs we are using are of sine and cosine of each quadrant:

Quadrant 1- Positive ÷ positive =positive

Tan,Cot,Sin,Cos fall in positive.

Quadrant 2- Negative ÷ positive=negative

Sin positive; Cos,Tan,Cot negative 

Quadrant 3- Negative ÷ negative= positive

Sin and Cos negative; Tan and Cot positive

Quadrant 4- Negative ÷positive=negative

Cos positive; Sin negative, Tan and Cot negative

The trick was used above. What it is is that the division we do from sin and cos gives us the sign (positive or negative) in which the  tangent and cotangent lie in.

The picture above shows sin.cos,tan.cot being graphed. You can see the trick visually on the graph. 

CSC/SEC

The reciprocal will lie in the same position to its corresponding trig ratio.

Quadrant 1-

Sin,Cos,Csc,Sec fall in positive

Quadrant 2-

 Sin positive Csc positive; Cos, Sec negative

Quadrant 3-

Sin,Cos,Csc,Sec negative

Quadrant 4-

Cos,Sec positive,Sin, Csc negative

Unit T Big Question#3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill? Use unit circle ratios to explain.

What is the concept about?

This concept is going to explain how tangent goes uphill and cotangent goes downhill. It will clarify why these two graphs have the different motion yet having the same pattern.

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

The viewer needs to pay attention to where the asymptotes are drawn for cotangent and tangent. This determines the way the graph will be shaped. They may have the same pattern, but the graphs are shifted differently when drawing them. We also need to remember the positive and negative pattern because we are going to use it.

Explanation:

There is a difference between the shapes of the graph because of  where the asymptotes are placed for tangent and cotangent. 

Tangent:

The tangent has the ratio identities of sin/cos. Since the asymptote is placed at pi/2 and 3pi/2 the graph is drawn going upward. The pattern is positive and negative, positive and negative. The negative is going upward to the next period which is positive, then the next period starts negative going up to positive. This gives it the uphill motion.

Cotangent:

The cotangent has the ratio identities of cos/sin. This means the asymptote are located in the 0 and pi. The pattern is positive and negative, positive and negative. This period starts at the positive and then goes down to the negative period. Then again it starts at the positive and goes down to the negative period. This gives it the downhill motion. 

Unit T Big Question #4

Why does sin and cosine NOT have asymptotes, but the other four trig graphs do? Use the unit circle ratios to explain?

Asymptote: A line that approaches the given curve but makes no contact with it. 

What is this concept about?

This concept explains why the cosine and sin do not have asymptotes and the other trig ratios do. We will be using ratio reciprocals and ratio identities from Unit Q to explain why trig ratios have asymptotes.

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

The viewer needs to pay attention to how each distinct place  is undefined on the graph, in other words where the asymptotes are placed when graphing. They also need to pay close attention to whether it is being drawn positive or negative in there period. Pay attention the explanation below on why sine and cosine don't have asymptotes.

Sine/Cosine

Sine does not have an asymptote because the ratio is y/r, R always equals one on the unit circle, therefore it can never be undefined. Same goes with Cosine it's ratio is x/r, the R is always one, so its not undefined. What both of these do have is an amplitude of one which was explained in Big Question #1 part B.

The pictures below explain how ratio identities and reciprocal identities are involved. It as well shows drawings of the asymptotes. 

Tangent

Tangent has a ratio identities of sin/cos, so whenever cosine equals zero the tangent is undefined leading it to have an asymptote. The asymptote on the graph is located wherever cosine equals 0 on the unit circle, in this case pi/2(90deg.) and 3pi/2(270deg.). 

   

Cotangent

Cotangent has the ratio identities of cos/sin, so wherever sine equals 0 the cotangent is undefined leading it to have an asymptote. The asymptote is located where the sin equals 0 , in this case at 0(0deg.) and pi(180deg.). 

Secant

Secant is the reciprocal of cosine, this is known from the reciprocal identities sec=1/cos. So similar to tangent whenever cosine equals 0 it is undefined leading it to an asymptote. The asymptote on the graph is located wherever cosine equals 0 on the unit circle, in this case pi/2(90deg.) and 3pi/2(270deg.). 

Cosecant

Cosecant is the reciprocal of sine and we know this by the reciprocal identities of csc=1/sin. Similar to cotangent whenever sin equals zero is where the asymptote is located.  The asymptote is located where the sin equals 0 , in this case at 0(0deg.) and pi(180deg.).  

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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Assessment # 3; #1 Unit S Concept 3

What  is the video about?
This is Assessment #3 Unit S Concept 3 being shown. This problem goes over #1 on the SSS packet, it explains power reducing formulas.  

What the viewer needs to pay close attention to in order to understand the concept?
What the viewer needs to pay close attention to is is what power reducing formulas we will be using in order to solve the problem. The main, very important  concept is that we need to remember our goal is to get the equation to the highest power of one. Another important key the viewer can pay close attention to is using the tactic from Unit Concept 4, with using the M; pay closes attention to how this is explained because it can be tricky. 

Assessment #2; Comparing Half- Angle & Sum/Difference

We know the answers are the same on both because we confirm it with our calculator. We plug each equation into our calculator and we should get a decimal point. Then we enter the 105 degrees with sine, cosine, and tangent and receive and equivalent decimal from the ones in the  equation. What I got for this particular equation is : Cosine= -.258, Sine= .965, Tangent= -3.732.Thank You for viewing!!

Assessment #4; Unit S Concept 7 #5

What is this problem about?
The problem is from Unit S Concept 7, solving equations with half-angle formulas. It shows us how to get our answer with replacing equation with half angle formulas. 

What the viewer needs to pay close attention to in order to understand the concept?
The viewer needs to pay close attention to what formulas we use in order to replace certain parts of the equation. There is a Pythagorean identity being used, we learned this from previous unit. We also need to use the unit circle to get our exact answers. Thank You for viewing!!