BQ#4: Unit T Concept 3

Tangent & Cotamgent:
A "normal" Tangent graph will go uphill because of the unit circle ratio: y/x. In Tangent's case, the x is what affects the placement of the graph and it's asymptotes. For a Cotangent graph, it also follows the unit circle ratio of x/y. However, for a Cotangent graph the y value affects the shifts of the graph causing it to to go downhill. The asymptote placement is the only factor that determines the direction of the graph. A normal tangent graph is uphill and a normal cotangent graph is downhill. The asymptotes for the graphs are in the same place, however cotangent has shifts.

BQ#3: Unit T Concepts 1-3

A) Tangent
Tangent is related to sine and cosine graphs because of the identities from Unit Q, and because of tangents ratio. As we previously learned, Tan=sinx/cosx which means that an asymptote would appear when cosine is equal to 0. The ratio from the unit circle for tangent is y/x, which also applies if x equals 0 then it will be undefined and have asymptotes. Depending on the sign of each x and y value will determine the direction of the graph.

B) Cotangent
Cotangent is the reciprocal of tangent which makes it also related with sine and cosine except the ratio is x/y. In this case, when y is equal to 0 there will be asymptotes. The direction of the graph depends on the value of sine (x).

C) Secant
Secant is the reciprocal of cosine which makes if closely related to cosine. The ratio for secant is r/x. The sign of x is the determining factor for the asymptotes. We know the right sign for x when we do ASTC per the quadrants in the unit circle.

D)Cosecant
Cosecant is the reciprocal of sine which makes the ratio r/y. Since r is always equal to one, the determining factor for the asymptotes is the y value of the graph. The graphs never touch the asymptotes but they come very close just as in every other graph.

BQ#5: Unit T Concepts 1-3

Sine and cosine will never have asymptotes because they will never be undefined. Since their ratios are y/r and x/r with r being 1, there will never be an undefined answer regardless if the value of x or y in the ratio.  However, tangent, csc, sec, and cot can be undefined which means they can have asymptotes based on their ratios: y/x, r/x, r/y, and x/y. If for example we were given tan with y=7 and x=0, we would get an undefined answer.

BQ #2: Unit T Concept Intro

Trig graphs relate to the Unit Circle in the sense if ASTC, they have the same signs that each trig function has according to the Unit Circle. A period for sine and cosine is 2pi because it has to go around the entire thing to complete one period. It has to go around a whole time until it repeats it's pattern again. With a tangent and coltan gent graph, it only has to go half way around before it repeats it's pattern which makes it only pi. The biggest values for a sine and cosine graph is 1 or -1 which sets the boundaries for its amplitude. This is based off of the trig ratios for sine and cosine in the Unit Circle. The ratio for sine is y/r and for cosine it's x/r, with the value of r always equaling to 1.

Reflection #1: Unit Q Verifying Trig Identities

1. When you are asked to verify a trig function that basically means that the terms you are given must in one form or another equal what is on the other side of the equal sign. Using identities to simplify the process with their substitution, you can get the identities to help you get alike terms on both sides of the equal sign to less complicate the verifying.
2. Some tips and tricks I have found helpful are taking a certain problem and splitting it into pieces rather than dealing with the whole problem at once. First I try to figure out if I have to substitute identities that will be similar throughout. If that doesn't seem to be an option then I multiply by the conjugate if I am given fractions. When all else fails I try the substituting identities and kind of play around with what I'm given until I get it to a simple enough route that I know I can solve.
3. When I first see a verifying trig problem I look at the terms given: sin,cos,tan,csc,cot,sec. Then I see if i have any identities that i can use as substitution for the problem. If I notice that the substitution of the identities complicates the number of steps to achieve the verifying then i retrace my steps and rethink my technique. Other options I have are probably dividing, adding, multiplying by the conjugate, or subtracting from one side to the other. I make sure to keep a close eye throughout my steps and make sure that there isn't an identity I can use whether it be a ratio, Pythagorean, or reciprocal. I don't have a very strategic technique other than trying different methods in order to make the verifying simpler. I have found that taking apart the problem helps visualize it clearer and allows you to focus on a particular situation instead of missing a step from dealing with the whole problem.