Big Question Blog Post Unit T Question 2

Question:
2. How do the graphs of sine and cosine relate to each of the others?
a. tangent?
b. cotangent?
c. secant?
d. cosecant?

Explanation: 
a. The sine and cosine graphs explained in the first concept relate to the tangent graphs by them dealing with writing the domain the same way and following almost the same process to finding the period shifts and the rest. The difference now is that we are dealing with figuring out the asymptotes in the graph due to the ratio of sin/cos where cosine for tangent at times will equal zero to give you an undefined answer which will lead you to having to draw asymptotes. 

b. For Cotangent graphs it is the same as tangent graphs except now the difference is that the ratio of a cotangent graph is different meaning the asymptotes will change. Now, the ratio is cos/sin where now sin becomes the one that is equal to zero at some point with the reference points meaning that asymptotes will have to be drawn and solved for.

c. The sine and cosine graphs are similar to the secant graphs in the sense that the secant graph follows the same process when you do cosine graphs except now you have to worry about asymptotes. Being that one of the values of the ratio of secant has a denominator of zero allows you to recognize that there will be use of an asymptote. Asymptotes are  use when you get an undefined answer and this is shown through secants ratio which is r/x where at one point x is equal to zero making the solution undefined because r always equals 1. So when drawing these asymptotes and labeling them on our graph we have to take a look at it and draw our parabolas that lie between these asymptotes. Also it can be explained as secant equalling 1/cos, where at one point cosine in that function will equal zero meaning that a asymptote will appear. 

d. As far as cosecant goes it also follows the same process as a sine graph yet now we are dealing with the ratio of r/y where y at one point is equal to zero. The denominator can never be zero but at one point with the given reference angle ordered pairs it is clear that the denominator (or x) will be zero which will give you an answer that is undefined. As stated before these undefined solutions allow you to plot the asymptotes and change the way the graphs are made because it now becomes a u shape (parabola). This one can also be explained as cosecant equalling the ratio of 1/sin. it is evident that sin will have to equal zero at one point which will make an asymptote appear. 

Big Post Blog Questions Unit T Question 4

Question 4:
Why do sine and cosine NOT have asymptotes, but the other four trig functions do?

Explanation:
Sine and cosine do not have asymptotes because as we know from previous units sine is equal to the ratio y over r and we know that r is always equal to 1 so with this knowledge it is easy to see how each ordered pair from each unit does not end up equalling undefined. From the points on the unit circle at each reference angle: (1,0), (0,1), (-1,0), (0,-1) if we plug in each of the y values into the ratio that is equal to sine we get real numbers such as 1, -1, and zero. When we plug in each of these values there are no equations that leave us with an answer as undefined such as tangent and cotangent do. As for cosine, it follows the same process when proving that there are no asymptotes but instead now the ratio we use is x over r where r still has the value of 1. When plugging in each of the ordered pairs x values it is evident that there are no undefined answers and just like sine your answers are either: 1,-1, or 0. There aren't any instances where the denominator is zero in when using these two ratios for sine and cosine. By noticing this we know that there will be no undefined answers meaning there will also be no asymptotes. As for the other four trig functions such as cotangent and tangent it is clear to see how there ratios (x/y and y/x) allow them to have an asymptote. They have an asymptote when the denominator of their ratio is equal to zero. Same goes for cosecant and secant graphs. 

Big Question Blog Posts Unit T Question 3

Question 3:
Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

Explanation:
This question can be explained by basically telling that the asymptotes of cotangent and tangent graphs differ in movement. They are divided separately because the signs of tangent don't start repeating their pattern until they go through each quadrant while cotangent just goes through the first two quadrants and its pattern repeats. By these two picture it is easily shown the difference, there are more asymptotes for a tangent graph than a cotangent graph due to the ratios use and their solutions of getting undefined. The ratios such as sin/cos=tan and cos/sin=cot are what seem to determine these asymptotes because the denominator cannot equal zero. The asymptotes and where they are drawn are basically what determines the difference in the movement of these two graphs. As seen in the picture for the cotangent graph there is an asymptote starting at 0 and another one at pi while tangent has its first asymptote starting at pi/2 which means that the curve lands in the first quadrant which is positive meaning  that the line is going uphill but then when it goes into the second quadrant tangent is negative in the second quadrant which makes sense that the graph go downward but upward again in the third quadrant.

Asymptote for cotangent graph

Asymptotes for tangent graph 

Citations:
Picture 1:
http://aventalearning.com/content168staging/2008Trigonometry/unit4/images/MTH08-68.20159.jpg
Picture Cot:
http://www.drdelmath.com/slu_precalculus/trig_images/trig_graph_cotangent.gif

Big Questions Blog Post Unit T

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?
b. How does the fact that  sine and cosine have amplitudes of one (and other trig functions don't have an amplitude) relate to what we know about the Unit Circle?

Answers:
1. The trig graphs relate to the unit circle by them following the same repetitive patter when going in a circle yet not it is shown as a line on the x axis. For example, for the trig graphs the quadrants used in the unit circle are now drawn on a line rather than a circle where the sign changes for each different trig function still remains the same.

                                            In this picture you can see how the renewed unit circle looks and how familiar it looks to the actual circle by containing the three major points (90, 180, 270, and 360) all on this axis which does not lose its significance when figuring out the signs of each trig function. The signs remain the same as when we figured them out using ASTC, if we even wanted to we could still write ASTC on it and still be able to use it. 

a. The period relates to the unit circle because one revolution around the unit circle is basically one cycle in which it takes for the pattern of signs and everything to repeat itself. So just like a period on these graphs it shows or lets us know how long it takes the trig function to go through one cycle. The period for sine and cosine are 2pi because thats how long it takes to go through one time of their cycle for it to then repeat itself again. They cover 2pi units on the x axis before repeating themselves all over. This picture shows just that, the pattern from 2pi repeating over again once going through a cycle. 

As for cotangent and tangent the period is pi because again that is how long of a cycle it takes for one cycle, before the pattern of these two repeat itself. They have to cover pi units in order to start repeating there cycle once again. 

b. The trig graphs relate to the amplitudes by them dealing with the numbers dealing with the coefficient that is connected with the trig function. The fact that sine and cosine have amplitudes relate to what we know about the unit circle because on the unit circle we are familiar that at each reference angle 0, 90, 180 and 270, they have points connected to them and when looking at the points at 0 is (0,1) and at 180 its (0,-1). These two points on the graph simply show how sine is 1 because it starts at 1 and ends at 1 for one cycle on the Unit Circle. As for cosine this is evident through the points at 90 and 270 where 90 has the points (1,0) and 270 has the points (-1,0), through this knowledge from the unit circle it is easy to see how both cosine and sine have amplitudes of one because it is dealing with the ordered pairs and the cycle needed for one of the trig functions. The other trig functions can also be explained to have no amplitude because if you think about it for tangent you would either be getting undefined or 0 as the amplitude, same goes for cotangent. 

Citations

Picture 1: 

http://mathbits.com/MathBits/StudentResources/GraphPaper/trigsmaller.jpg

Picture 2:

http://img.sparknotes.com/figures/A/ad79275cb59e569b790cb945a4ffc553/quadrantgraph.gif

Picture 3:

http://www.s-cool.co.uk/a-level/assets/learn_its/alevel/maths/trigonometry/graphs-of-trigonometric-functions/2007-10-09_133331.gif

Picture 4: 

http://www.calculatorsoup.com/images/trig_plots/graph_cot_pi.gif

Assessment #4: Unit S Concept 7

Unit S Concept 7 #4

• What is this problem about?

This video goes over a real problem from Unit S Concept 7. This concept is dealing with solving equations that contain half angle formulas that could be substituted into the equation in order to simplify it further. This problem makes use of previous concepts and  even previous units that deal with the unit circle radians. This concept allows us to research back into what we have learned earlier and just basically put it all to use. For example, we also make use of powering up in order to simplify further. Factoring is also something in which this problem entails for us to know in order to simplify and get the correct exact answer.

• What does this viewer need to pay special attention to in order to solve the problem correctly?

This viewer needs to pay special attention to the half angle formula we plug into this equation and why. They need to also pay attention to how that formula is dealt with in the problem, being that there is a radical and it may be confusing. The viewer also needs to be able to see how we were able to substitute in a variable for cosx in order to factor and get rid of the square. The process in solving this problem can get confusing so I suggest that the viewer pay special attention to the steps taken to first get the problem into a form that we can work with such as getting rid of the radical, squaring both sides, and factoring. 

Assessment #3 Unit S Concept 3

• What is this problem about?

This problem goes over a real problem from Unit S concept 4 which is dealing with using  power reducing formulas. The formulas used in this concept are just as it read, used to reduce the power of the original problem given so that our final product could all be in the first power. In this problem we deal with substitution, foiling, and with common denominators. It makes use of a lot of our algebra skills but still deals mainly with substitution by making use of these power reducing formulas. 

• What does this viewer need to pay special attention to in order to solve the problem correctly?

This viewer needs to pay special attention to how to use the power reducing formulas by breaking up the original equation if dealing with power greater than 2. After doing this the viewer needs to pay special attention to how we deal with the problem after the power reducing formula is substituted in. It starts getting a little difficult after we use m to equal a certain number and how we then plug the value back into with the rest of the equation. The simplification of the rest of the problem, I feel, is most confusing and most difficult to do because of all the substitution it entails. 

Assessment #2 Unit S Concept 4

THE UNIT S WAY ^

UNIT R WAY^

Describe how you know that those answers are the same. 

• When looking at how both these problems are solved differently, it seems as though the ending product is not going to equal the same value. After solving for each and simplifying we can see that one of the values from both ways match, that is the value of tangent. The other two trig functions are also proven to equal the same when you plug each into the calculator. When doing so you find that using both ways (sum-difference formula/half-angle formula) end up equalling the same for all trig functions. They end up having the same exact value even though they look different and are solved using different methods.