Unit T Big Question # 2

How do the graphs of sine and cosine relate to each of the others? 

a.) Tangent
b.)Cotangent
---The video will go over tangent and cotangent:



c.) Secant: Secant is the reciprocal of cosine. In order to graph the secant  graph we need to graph the cosine graph. Although cosine graphs do not have asymptotes secant graphs do so once we have draw the regular cosine graph we will then draw the asypmtotes for the secant graph. Secant is 1 / cos so the asymptote lies where cos is zero which is at pi/2 and 3 pi/2. So when we do draw the cosine graph with the secant asypmtotes the graph will look like this:

Now the only thing left to do is to draw the secant graph. Again secant is the reciprocal of cosine so its graph it going to be mirrored from it but as we do this we have to remember that the asymptotes are the barriers to our graph:(secant is represented by the green)

d.)Cosecant: For cosecant we follow the same concepts of that of cosine. First we draw the sine graph one a regular unit circle scale. Although sine does not have an asymptote, cosecant graphs do. Cosecant is 1/sin so whenever sine equals 0 there will be an asymptote since it will be undefined. This happens at 0, pi, and 2pi. So when we draw the sine graph with the cosecant asymptotes the graph will look like this:

Now the draw cosecant we see that there are two sets asymptotes so two graphs will be drawn. Again since  cosecant is the inverse of sine then the cosecant graph will be a reflection of the sine graph. Now when doing so we have to take two things into consideration:

                           *the sine graph was divided into two sections because of the asymptoes so that means there will be one cosecant graph in between each of the asymptotes for a total of two cosecant graphs. 

                         *Between 0 and pi cosecant is positive so the graph has to be reflected above the x-axis and cosecant is negative between pi and 2 pi so the graph needs to be reflected below the x-axis. 

The graph ends up looking like this:

                                                    

Citations for images:

http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphtrigfunctions.html
http://www.intmath.com/trigonometric-graphs/1-graphs-sine-cosine-amplitude.php

Unit T Big Question # 3

Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill? 

• Looking at the Unit Circle and ASTC, we see that tangent is positive in the first quadrant, negative in the second, positive in the third, and negative in the fourth. Now if we take the unit circle and make it so that it is on a graph it would be like this: 

Tangent is sine over cosine and we know that there is an asymptote whenever the value is undefined. Since tangent is sin/cos then the value will be undefined whenever cosine equals zero and that happens at pi/2 and 3 pi/2. 

                                                 pi/2 ( 0, 1) -------- y/x= 1/0=undefined

                                                 3 pi/2 (0,-1)-------y/x= -1/0= undefined

So we have an asymptote at pi/ 2 and 3 pi/2 which is represented by the picture below:

Now that we have the asymptotes we have the barrier to each graph. So we had said that tangent is positive in the first quadrant so that means that between 0 and pi/2 the tangent curve will be going up above the x-axis towards the asymptotes but not touching it. Between pi/2 and pi tangent is negative so the curve will be below the x-axis going up towards the asymptote. If we do this for all the quadrants we get a result of the following graph, which shoes the final tangent graph:

* This is why the tangent graph goes up and it is really based on the placement of the asymptotes that guide how we draw tangent graphs. 

• For cotangent the same process is taken to explain why cotangent graphs are downhill.

 Looking at the unit circle cotangent is positive in the first quadrant, negative in the second, positive in the third, and negative in the fourth. So again like tangent we draw the unit circle on a plane. This type however cotangent is cosine over sine. So this time the asymptotes will lie whenever x/y is undefined.This only happens at pi and 0 or 2 pi.

                                               pi ( -1,0)------x/y= -1/0 = undefined

                                              0 / 2 pi (1,0)------x/y= 1/0= undefined 

*So we have asymptotes at pi and 0 and 2 pi. 

Now we set our barriers we can draw the graph. Between 0 and pi/2 cotangent is positive so it will be above the x-axis only this time the curve will be going down because the it is framed by the asymptote. We do this for all the quadrants and in result get the following graph,which shows the final cotangent graph:

Citations for images:

http://www.biology.arizona.edu/biomath/tutorials/trigonometric/graphtrigfunctions.html

http://demo.activemath.org/ActiveMath2/search/show.cmd?id=mbase://LeAM_calculus/curves/ex_cot_pole

Unit T Big Question # 4

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? 

Watch the video and find out why!! :D

Unit T-Big Question #1

How do the trig graphs relate to the Unit Circle.?

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

Trig graphs are clycical, meaning that they repeat themselves over and over gain. One time through this cycle is called a period. Looking at the Unit Circle we can divide into four quadrants in which each particular quadrant has certain trig functions that are positive or negative depending n the quadrant. We represent this by ASTC, like the picture below.:
                                 

-Sine: Looking at sine we see that sine is positive in the first quadrant, positive in the second quadrant, negative in the third, and negative in the third. In result, the pattern for sine is +, +, -, -. so we see that for the pattern to repeat itself again we have to go through one whole cycle around the unit circle thus 2pi since the untie circle is 2pi all the way around. The following picture shows how the pattern looks like on the unit circle:

-Cosine: Looking at cosine we see that it is positive in the first quadrant, negative in the second, negative in the third, and positive in the fourth. The pattern is +,-, -, +. Like sine we see that it takes one whole cycle around the unit circle for the pattern of cosine to repeat itself so that is why the period is 2pi. The following picture shows how the pattern looks like on the unit circle:

            *Remember a period is when the pattern starts and end. Because it takes both sine and cosine one whole cycle around the unit circle to repeat themselves the period is 2pi. 

-Tangent/Cotangent: Looking at cotangent and tangent both are positive in the first quadrant, negative in the second, positive in the third, and negative in the fourth. The pattern for cotangent and tangent is +, -, +, -. We can see that it takes half the time of sine and cosine for the pattern of tangent/cotangent to repeat itself. We have +, -, and then again +, -., so the pattern begins to repeat itself half way through the unit circle which is pi. The following picture shows how the pattern looks like on the unit circle:

b.) Amplitude? How does the fact that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?

 --Looking at the Unit Circle we see that the greatest distance we have is one. This puts a limit at to how far the sine and cosine graphs can reach thus making it its amplitude.  The sine of 90 degrees is 1 and the sine of 270 degrees is -1, which further explains that the amplitude is 1. The cosine of 0 degrees is 1 and the cosine of 180 degrees is -1 which again makes the amplitude 1.


citations for images:

http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceTriangles.htm

http://etc.usf.edu/clipart/galleries/617-unit-circles

Unit S Assessment #3

Unit S Concept 3







What is this about?
A problem is presented from Unit S Concept 3 which is using power-reducing formulas. This video goes over one of the problems from the pq's and goes over s step by step explanation on how to solve it. It shows how a function that has a power of four can be reduced to a function that has a power of one using various identities and formulas. It involves product-to-sum formulas and much more.

What does the viewer need to pay special attention to in order to understand the concept?
- One thing that the viewer need to pay close attention to is how the constant "m" was used and how it was substituted back. There is a lot of simplifying in this problem so it is important to carefully substitute it where it belongs. Just like when we use "M" and then we have to go back and place back in what "m" equals and so on.

- Another important to pay close attention to is the various formulas that were used. While simplifying the problem we see a lot of identities that we can use and go further to plugging it into the formula they correspond to. When doing so it important that the viewer is able to see why the certain formula was used and how it was plugged in.

Unit S Assessment #2

Unit S Assessment 2

How are these the same?
We know that these two are the same because for both problems are the same and we are looking for the same thing in both problems. In both pictures we are solving for the sin, cosine, and tangent for the angle value of 75 degrees only that they are solved in different methods. In the first picture we solved using sum and difference formulas and in the second picture we solved using half-angle formulas. We can also verify that these two are same by plugging in the values and solving using the calculator. We would get the same sine value for both results and so on.

Unit S Assessment #4

Unit S Concept 7

What is this about?

A problem is presented from Unit S concept, which is solving equations with half-angle formulas. This problem goes over solving an a trig equation that contains a half-angle using the half-angle formulas learned in this unit. Knowledge from previous units are needed to solve this problem. Things such as the unit circle values, identities, and extraneous solutions will be dealt with in this problem.

What does the viewer need to pay special attention to in order to understand the concept?

-The viewer need to pay close attention to is to the identities being used in this problem especially the ones reviewed from unit Q and how they are being substituted in. 

- The second thing to keep in mind is that at the beginning of the problem we squared the equation and since we raised the problem it is very important to go back and check for any extraneous solutions.