BQ#3: Unit T Concepts 1-3: Trig Graphs

How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response. 
-Tangent
-Cotangent
-Secant
-Cosecant 

1) Tangent 
     Okay so the ratio identity for tangent we know is sine/cosine. Also, we are aware that we will only have asymptotes where we get an undefined answer meaning that we divide by zero and therefore, cosine would have to be zero in order to make this asymptotes and an undefined answer. Now, we will have asymptotes at the 90 and 270 degree, so at pi/2 and 3pi/2. This being said, we know that tangent cannot touch them no matter what. Now, there are way more asymptotes, they practically go on forever, yet we can just stick to some.

2) Cotangent 
        We can basically use the same knowledge from above with Tangent, yet now we know that cotangent is equaled to cosine over sine and so the asymptotes will be found where cosine is zero so at 90 and 270 degrees, or pi/2 and 3pi/2. And we remember that the graph cannot touch the asymptotes!

3) Secant 

      We know that secant is the reciprocal of cosine and the ratio is 1/cos. Secant as well has asymptotes that are found where cosine is equaled to zero as well. 

4) Cosecant 

      Now, the same rules apply like with the reciprocals and stuff yet the asymptotes will be located in different places due to the different kind of graph. Cosecant is the reciprocal of sine so its ratio is 1/sin. 

References:

http://mathsm1m0838.edublogs.org/files/2009/08/Picture-11.png

http://2.bp.blogspot.com/-KGMdRWBNlP8/UXi7SKh2BjI/AAAAAAAAAEU/YSex9MpWZAk/s640/2013-04-24+22.05.53-1.jpg

http://www.mathipedia.com/GraphingSecant,Cosecant,andCotangent_files/image014.jpg

http://www.ask-math.com/images/Cosecantgraph.png

BQ#4: Unit T Concept 3: Tangent and Cotangent Graphs

Why is the "normal" tangent graph uphill, but a "normal" cotangent graph downhill? Use unit circle ratios to explain.
        
        The reason that a tangent graph is uphill and yet a cotangent graph is downhill is due to the fact that they are different ratios and different of which then creates different asymptotes. We know that tan=sin/cos and cot=cos/sin, or in other words, tan is equaled to y/x and cot is equaled to x/y. See how the denominators are different, tangent has a denominator of cosine while cotangent has a denominator of sine and therefore for tangent we would get our asymptotes wherever cosine is equaled to zero which would be at the 90 degree (pi/2) and at the 270 degree (3pi/2). Furthermore, for cotangent we would get asymptotes wherever sine is equaled to zero which would be at the 0 degree (0) and at the 180 degree (pi). We do know that the pattern for both is positive, negative, positive, negative, yet the asymptotes are located in distinct places and so creates a different looking graph.   

References:

-http://www.mathamazement.com/images/Pre-Calculus/04_Trigonometric-Functions/04_06_Graphs-of-Other-Trig-Functions/cotangent-graph.JPG

-http://www.regentsprep.org/Regents/math/algtrig/ATT7/otherg91.gif

BQ#5: Unit T Concepts 1-3

Why do sine and cosine NOT have asymptotes, but the other four trig graphs do? Use unit circle ratios to explain. 
     The reason that sine and cosine cannot have asymptotes is due to the fact that they will never be undefined. Now, let's recall that in order for there to any asymptotes then we have to have an answer as undefined. Okay, with that in mind we can now prove that sine and cosine do NOT have any asymptotes. We know that the ratio for sine is y over r and as for cosine, we know that it is x over r. "R" is always equaled to one right? And since we will never have a zero be on the denominator, then that means that no asymptotes will be present and created. 
      
     The other four trig graphs do have asymptotes and that is because they will not ever have r or a 1 as the denominator. For example, if we did tangent, then we know the ratio is sine (y) over cosine (x) and we know that x can be a zero at the 90 degrees and at the 270 degrees, or pi and 3pi over 2. Also for secant, it would be the same thing since the ratio is r over x and x can be a zero like stated before. Lastly, cosecant (r/y) and cotangent (x/y), they have a denominator of y and we know that y can be equaled to zero at 0 degrees and 180 degrees, or in other words, 0 and pi.  

BQ#2: Unit T Concept Intro: Trig Graphs and the Unit Circle

How do trig graphs relate to the Unit Circle?
   Okay so I know what you are thinking..the Unit Circle again?! Well yes, the Unit Circle comes in from everywhere and somehow has a function for everything. For trig graphs, it is as if you are seeing the Unit Circle, yet in a straight line and the same numbers are present in order. Also, their signs depend on what quadrant they are found in. For example, if we have cosine, we know that it is positive in quadrants 1 and 4.

1. Period? Why is the period for sine and cosine 2pi, whereas the period for the period for tangent and cotangent is pi?
    The period for sine and cosine is 2pi due to the pattern that it follows and how it repeats after every four marks on a graph. Now, for tangent and cotangent and we are able to see that it repeats every two marks. For proof we can see it after knowing that the symbols are positive, negative, positive, negative and so that means that it only takes two marks. 

2. 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 first of all, what are amplitudes? Well an amplitude is half the distance between the highest and lowest points on the graph. They can be found by looking at the value of |a|. Sine and cosine have the same amplitude of which is 1 and -1 due to the fact that they are the ones with the restrictions. They cannot be any points greater than or less than 1 and -1 since they don't exist anymore. Now, we get to tangent and cotangent. They have no restrictions and therefore can keep going on and on and on. Their range will always be negative infinity to positive infinity. Lastly, we have cosecant and secant of which simply don't have a range and therefore no amplitudes. 

Reflection#1: Unit Q: Verifying Trig Functions

Dear Future Math Students,

Unit Q deals with several fundamental identities, trig functions, ratios, and using different simplification processes in order to solve and verify as well. I know that this sounds like gibberish and that you probably want to cry because it just seems so confusing and you just do not want to go through this, but DON'T WORRY! I did it and it was okay....after you practice practice practice and get the feel for it. I am not going to lie, IT IS DIFFICULT! But it will not be the end of the world! Do not give up and try and practice and sooner or later you will succeed! (: Okay well let me go more in depth...


1. What does it actually mean to verify a trig identity?
-To verify simply means to check and prove that whatever is stated is true. For example, if you had 1+1=2 then you simply have to go through steps that will help you prove that 1+1 actually does equal 2. Of course, it will not be a nice and easy as my example. ...Sorry ._. And so, basically that you have to be sure to solve a problem and get the same answer as the one that is given and equaled to. Oh! and it is better when you have to verify rather than having to just solve and come up with an answer. The reason is that when you are verifying, you know the answer you should end up with and it is just you having to plug in and substitute and solve until you hit bingo and check the answer, whereas when it comes to just solving and you are obviously not given an answer, then you just have to solve correctly and pray you get it correct.

2. What tips and tricks have I found helpful?
-Okay so we know that there are a bunch of functions and ratios like sine, cosine, tangent, cosecant, secant, and cotangent, yet a tip I have is to turn everything into sin/cos at some point because it will become easier. Sine and cosine can be considered our best friends in this case. They are nice to us, so might as well use them. (; Also, I have foreseen the future and I know that you will feel discouraged and defeated, but NO! You can do this! DON'T GIVE UP! This is a tip that I know you WILL need! So use it! Keep working through it and take various adventures with the problem.

3. Explaining my thought process and steps I take in verifying a trig identity.
-Okay so if you noticed, I said that you should take various adventures with the problem and what did I mean? Well the beauty of these problems is that there isn't only one way on solving a problem so you really can't o wrong! Yay! If you begin differently than someone else, it's okay just keep going and don't get stuck, use your knowledge, you identities, your reciprocals. Now, when I encounter a problem I think and wonder what I should do first. Should I turn into sine and cosine? Should I multiply by the conjugate? Should I split the problem up maybe to see if something cancels? Or maybe I should find the least common denominator? Perhaps this is an identity? Honestly, I just think of what to do first and then I keep going just trying stuff and see where it takes me and I just use the skills I know. There is never one path so don't be afraid of trying something! (:

SP#7: Unit Q Concept 2: Finding all trig functions when given one trig function and quadrant (using identities)

"This SP#7 was made in collaboration with Tracey P. Please visit the other awesome posts on their blog by going here."

Now, there is more than one way to find our answers and solve for the function, two are those that will be expressed throughout this post. We know that we can solve this using identities or by using SOH CAH TOA! So, the following pictures will display the solutions to the problems in both ways and using both methods. (: In the end we will find out how you get the same answers either way!

The first image will show and go over how to solve this function when using identities.

This second image will go over solving the function by using the wonderful SOH CAH TOA!  

In order to be able and understand this problem and be able to deal with it as you solve it, you have to be able to know the different types of identities and they should be able to mix and match and substitute several terms for others. Also, of course you have to know what SOH CAH TOA is and the ratios for each. Also, you should be familiar with all of the ratios and how there could be more than one. It can be difficult, but with practice everything will be fine. (:  

I/D#3: Unit Q Concept 1: Pythagorean Identities

Inquiry Activity Summary:
1. Okay so we know that we are dealing with identities because the title is a given, yet what are identities? Well, an identity is basically any proven fact or formula of which brings the Pythagorean Theorem to mind considering that it is a formula that works and is always correct when put to use. Okay so let's start off with the Pythagorean Theorem. If we were given the terms x, y, and r then we would write out the formula like this: x^2+y^2=r^2, but we would tweak it a bit in order to have it equal to 1. We would divide by r^2, do we agree? Yes! Because that is pretty much the only way to equal an equation to 1, by dividing by whatever number the equation is equaled to. Then we would have (x/r)^2+(y/r)^2=1.
 
     Now, if we go back to the unit circle, oh yeah I forgot to mention that the unit circle does come back to haunt haha, we know that cosine is x/r or in other words, adjacent over hypotenuse and sine is y/r or opposite over hypotenuse, so ..do we see anything? Hmm.. Yea! It matches to much of the Pythagorean Theorem! We would just have to square the two variables on the left, x and y, and then equal them to 1. In the end we could have cos^2theta + sin^2theta= 1. Remember that cosine is x and sin is y so in reality we just do some switching around and have fun with it and therefore sin^2x+cos^2x=1 is referred to as a Pythagorean Identity. To prove the theory with some examples and so you can see it visually look at the images below if let's say we did the magic 45 degree.

2. Now, to derive the identity with Secant and Tangent we would start off with the original sin^2x+cos^2x=1. The first thing we would do is divide everything by cos^2x and we would get sin^2x/cos^2x + cos^2x/cos^2x= 1/cps^2x and we can still simplify this further! We know that sin^2x/cos^2x is equal to tan^2x because tangent is opposite over adjacent (y/x). For the easy one, we know that anything over itself is 1 so cos^2x/cos^2x is 1 and then lastly we know that 1/cos^2x is the same thing as saying sec^2x. We simply are plugging in until it simplifies nicely. To conclude we have tan^2x+1=sec^2x. Yay! 

    To derive the identity with Co-secant and Cotangent we would start with the same original one, but divide by sin^2x. We know that after doing so the sines will equal 1 and cos^2x over sin^2x is the same thing as saying cot^2x so we would have 1+cot^2x= ...csc^2x since 1/sin^2x is csc^2x. 

    In order to see what is going on the picture below will hopefully help. 

 

Inquiry Activity Reflection: 

1. "The connection that I see between units N, O, P, and Q so far is...that everything in some sort of way comes together to conclude and involve the unit circle and triangles are involved as well. 

2. "If I had to describe trigonometry in 3 words, they would be...ratios, substitutions, and strangely comprehensible.