BQ #6: Unit U Concepts 1-8

1- What is a continuity? What is a discontinuity?
 A continuity is when the function is predictable, has no breaks, holes, or jumps, and can be drawn without lifting your pencil. A discontinuity is when the function DOES have holes, breaks, or jumps, and it isn't predictable. As well does a discontinuity have two types of families:
The first type of family is the Removable Discontinuities- This family only has the Point Discontinuity, which is pretty much a graphed out line with a hole in it (it can also be a hole with a point above it).
 Ex.

http://bfreshrize.files.wordpress.com/2012/01/unknown.jpeg?w=500

The second type of family is the Non-Removable Discontinuities- This family has three discontinuities. The first one is the Jump Discontinuity, which is similar to piecewise graphs however both parts cannot have closed points but can have them both open or one open and one closed.
Ex.

http://image.tutorvista.com/content/feed/u364/discontin.GIF

The second one is Oscillating Behavior, which is just a line graphed out that is extremely "wiggly" that it makes it almost impossible to find the height, limit, and value of it.
Ex.

http://webpages.charter.net/mwhitneyshhs/calculus/limits/limit-graph8.jpg

The last one is the Infinite Discontinuity, which is also known as unbounded behavior and it occurs were there is a vertical asymptote.
Ex.

http://www.math.brown.edu/utra/discontinuities%201.GIF

2- What is a limit? When does a limit exist? What is the difference between a limit and a value?
 A limit is the intended height of a function. The limit exists whenever you reach the same height from both the left and right (on a continuity). However it does not exist when you cannot reach the same height from both the left and right (on a discontinuity). The difference between the two is that a limit is the intended height while the value is the actual height of the function.
3- How do we evaluate limits numerically, graphically, and algebraically?
When evaluating limits numerically you just have to create a table to find the points closest to the value and vise versa to find the limit.
Ex.

http://media.showme.com/files/349881/pictures/thumbs/658760/last_thumb1359484583_200x150.jpg

To find limits graphically, we use our fingertips and follow the line to the point that we are looking for.
Ex.

http://www.mathnstuff.com/math/spoken/here/2class/420/c42d.gif

When finding limits algebraically, we use three methods: Direct Substitution, Dividing out/Factoring, or Rationalizing/Conjugate. However when finding limits algebraically we must always start with direct substitution to make sure that it equals 0/0 for we can do anymore, for if it equaled a real number to begin with all of the work done would have wasted time.
Ex. (Substitution)

Ex. (Factoring)

Ex. (Rationalizing)

BQ #3: Unit T Concepts 1-3

1-How do the graphs of sine and cosine relate to each of the others? Emphasize asymptotes in your response.
A: Tangent?

B: Cotangent?

C: Secant?

D: Cosecant?

BQ #4: Unit t Concept 3

1-Why is a "normal" tangent graph uphill, but a "normal cotangent graph downhill? Use the Unit Circle Ratios to explain.
The reason why is because there is a difference in the asymptotes. The asymptotes are shifted in different places because of the fact that tangent and cotangent are reciprocals, for example tan=y/x therefore making the focus be on "x" because when finding the asymptote "x" must equal zero and so the only areas that are equal to zero is at 90 and 270 degrees. While for cotangent, cot=x/y so the "y" id now the one that has to be equal to zero and that is at 0, 180, and 360 degrees, with that there is a shift from where the asymptotes will be at. Making tangent go uphill and cotangent go downhill. If you are still confused look at the picture at the bottom.

BQ #5: Unit T Concepts 1-3

1-Why do sine and cosine NOT have asymptotes, but the other four trig functions do? Use Unit Cirlce ratios to explain.
The reason why sine and cosine do not have asymptotes is because their trig ratios (sin=y/r and cos=x/r) and always on top of 1 ("r") making them to never be undefined. While the other four trig functions (csc=r/y, sec=r/x, tan=y/x, and cot=x/y) are not over 1 and therefore their denominator can equal 0 which will equal to be undefined and will create the asymptotes.

BQ #2: Unit T Intro

1-How do the trig graphs relate to the Unit Circle?
A) Period?-Why is the period for sine and cosine 2 pi, whereas the period for tangent and cotangent is pi?
The reason why the period for sine and cosine is 2 pi is because in the unit circle it takes them the whole 360 degrees in order to repeat it's pattern. For example for sine in the first and second quadrants it is positive but in the third and fourth quadrants it is negative so in order to see the pattern repeat one must wait until another 2 pi goes around. While for tangent it only takes pi because in the unit circle it is positive in the first quadrant and negative in the second allowing the pattern to repeat itself again in the next two quadrants. If unclear (for visual) look at the picture below.

B) Amplitude?-How does the fact of that sine and cosine have amplitudes of one relate to what we know about the Unit Circle?
Sine and cosine equaling to one is related to the unit circle because they both can be between 1 and -1 because of their trig functions we used in the Unit Circle before (sin=y/r and cos=x/r). While the other functions don't have amplitudes because of the fact that they don't have restriction. For example, tan=y/x because of this tan can be equal to anything that can be inside and outside the Unit Circle.

Reflection #1: Unit Q: Verifying Trig Identities

1- What does it actually mean to verify a trig identity?
To verify a trig function actually means to solve a very "complicated" equation with trig functions in them. the reason in why i say that they are "complicated" is because in all reality when you know how to do them, the problem will already give you an answer and you just have to see if it is true or you are to just simplify it to the best of your abilities.

2- What tricks and tips have you found helpful?
I found the tip of doing the hardest side first to be very helpful because when I finished the hardest I knew that I was able to solve the rest of the equation fairly quick because I already did the most challenging part of the whole equation. As well was the trick of memorizing the trig identities helped me a lot, yes it was a pain to memorize more equations however in the long run it helped me so much especially during the test because I was able to name identities and was able to cut the work load almost in half. As well as being to split fractions into two monomial denominators because it helped me a lot when I got stuck in solving equations that I thought might have not even had an answer.

3- Explain your thought process and steps you take in verifying a trig identity
Well my first step in doing this is to see whether there is an identity that I can switch for one of the Pythagorean theorems, but if there isn't any then I convert everything to sine and cosine. What I then do is try to cross cancel or I keep looking if there are more Pythagorean theorems. If not and they are in a fraction, then I split them into two monomial denominators or I use their reciprocals to cancel out and find Pythagorean theorems (I only use this when I feel like I am stuck on the problem).

SP #7: Unit Q Concept 2- Find all trig functions given one trig function and a quadrant

In this concept we see how we can find all trig functions when given just one trig function and a quadrant. You can find this very easily with SOHCAHTOA or just use this method to check your work. This is the problem of my partner, Genesis R., and mine. Please try to answer it before looking at the answers below:
Tanx= -8/5, Cosx= 5rad89/89

However this isn't the only way to find these answers if you want to learn how to get the same result but with Identities then click here to see how it is done.