BQ #4: Unit T Concept 3

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

Basically, a normal tangent graph is uphill because it is based on its asymptotes. They are located at pi/2 and 3pi/2. Furthermore, the quadrants are shown as positive, negative, positive, and negative (based on sin/cos). Cotangent is downhill because its asymptotes are located at 0 and pi. Because of where the asymptotes are located, it is downhill. The quadrants are positive, negative, positive, negative. The asymptotes are where sine and cosine are equal to 0 (making the ratio undefined).


                          Visual of Tangent 

(https://www.desmos.com/calculator)

As you can see, the colors represent the different quadrants (I,II,III,IV)

http://mathsm1m0838.edublogs.org/2009/08/01/the-tangent-graph/

Asymptotes at pi over 2 and 3pi over 2

  Visual of Cotangent

(https://www.desmos.com/calculator)

                                           

http://www.oocities.org/mathnuts/TrigAGModule1-TrigFunctions/graphs-general-tan-cot-sec-csc-sec6p8/graphs-general-tan-cot-sec-csc.htm

Asymptotes at 0 radians and pi

BQ#3 – Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each of the others?  Emphasize asymptotes in your response.

Tangent
Sine and cosine is related to tangent because the ratio identity for tangent is (sin/cos). The quadrants in the Unit Circle is shown in the tangent graph. Using our previous knowledge of the unit circle, we know where sine and cos is positive or negative in the quadrants.  If sine or cosine is negative, then tangent would be negative. If sine or cosine was positive or negative, tangent would be positive. In the first quadrant, sine and cosine are positive, so tangent is positive. In the second quadrant, cosine is negative, but sine is positive. But, overall tan is negative because cosine is negative. In the third quadrant, both sine and cosine are negative, making tangent positive (the negatives cancel out). The fourth quadrant for tan is negative because cosine is positive, but sine is negative, making tan negative. Tangent has asymptotes when cosine is equal to 0. Therefore, at pi/2 and 3pi/2, there are asymptotes. The asmymptotes repeat itself over and over again, as the domain goes on.

Visual of Tangent

(https://www.desmos.com/calculator)

Cotangent

In cotangent, the ratio identity is cosine/sine. It is very similar to tangent. Cosine and sine are positive in the first quadrant, so cotangent is positive. In the second section, the sine is positive and cosine is negative, so cotangent is negative. For the third quadrant, cosine and sine are negative, making cotangent positive (cancels out).  For the fourth quadrant, cos is positive and sine is negative, making cotangent negative. Sine must equal to 0 to find the asymptotes. Remember, asymptotes are undefined values. Therefore, we know that sine is equal to 0 are (1,0) and (-1,0). That is at 0 and pi. Cotangent is different from tangent because of the asymptotes. 

Visual of Cotangent

https://www.desmos.com/calculator)

Secant 

The reciprocal of secant is cosine. So the graph is just cosine flipped. So since cosine is positive in the first and last quadrant, and negative in the second and third, so is secant.  If cosine is positive, so is secant. Sine has no effect on the graph because it is not part of the ratio. The asymptotes are (0,1) which is pi over 2 and (0,-1) and 3pi over 2. This is where cosine is 0, making the ratio undefined. This is similar to the tangent graph, which had cosine as the denominator. 

Visual of Secant

(https://www.desmos.com/calculator)

Cosecant

Cosecant is just secant reversed. The ratio for cosecant is 1/sin. So basically, since sin is positive. so is cosecant in the first quadrant. In the second quadrant, cosecant is positive as well. In the third and fourth, cosecant is negative. The asympytotes are where sine is equal to 0 which is at (1,0) and (-1,0). Therefore, it is at 0 and pi. Remember, the graphs don't EVER touch the asymptote. 

Visual of Cosecant 

(https://www.desmos.com/calculator

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.
          Sine and cosine do not have asymptotes because they are never undefined. Remember, sine means y/r and cosine means x/r. R stands for the hypotenuse, and in the unit circle the radius is always 1, which is the hypotenuse. The other four trig graphs do have asmpytotes because they do not have 1 (r) as their denominator. Instead, their denominators can be any numbers, according to their trig ratio. For example, tangent's ratio y/x. X can be any number; there are no restrictions.
        The denominator, however, can be 0 (from the coordinates (0,1) or (-1,0), which makes the ratio undefined once in the according trig function. In cotangent, the ratio x/y also can assume asymptotes. If the coordinate is at (-1,0) or (1,0), when plugged into the ratio, it will be either positive or negative 1 over 0, which is undefined. Also, this applies to secant. Secant is r/y. Y is 0 at (1,0) or (-1,0). This results to be 1 over 0, which makes it undefined. In cosecant, the ratio is r/x. X is 0 as (0,1) and (0,-1). Thus, when plugged in, the ratio is 1/0, making is undefined as well. An asymptote is basically when the ratio is divided by 0, meaning its undefined. 

BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the unit circle?
So, you may ask, how do trig graphs relate to the unit circle? Well, it uses the positive and negative signs of the quadrants of the trig function! The trig graphs are basically the unit circle unwrapped or uncoiled to a line. The points on the graph are relevant to the points in the unit circle (radians). FOR EXAMPLE, in the unit circle, sine is positive in the first two quadrants, and negative in the third and fourth concepts (we know this from our previous units).  Thus, on a trig graph, the graph will show a pattern of going uphill, uphill, downhill, downhill. This represents the quadrants. It is positive in the first two quadrants and negative in the other two. Lets use another example. For cosine, in the unit circle it is positive in the first quadrant, negative in the second, negative in the third, and positive in the fourth. So on the trig graph, the period is positive and then goes negative, and eventually goes back positive according to the marks. For tangent and cotangent, it will have positive values first, then negative, then positive, then negative. The trig graph has a shorter period and this is because in the first two quadrants, it goes positive then negative, and then it is basically repeated in the other two quadrants. 

Beautiful visuals of the trig graphs and the patterns that I talked about in my earlier paragraph.

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

The period for sine and cosine is 2pi because basically when you look at the unit circle, sine is positive, positive, negative, negative on the quadrants. In cosine, it is positive, negative. negative, positive. The period is 2pi and it takes all the quadrants (one whole revolution) to finish the pattern. But, for tangent and cotangent the pattern is positive negative. If it is positive and negative in the first two quadrants, and it repeats after that, then basically it only takes the period of pi in a revolution, in which it repeats itself twice.

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?

Sine and cosine have amplitudes of one. Why? Well think about what sine and cosine consists of. Sine is y/r and cosine is x/r. R is the hypotenuse, which is 1. Sine and cosine cannot be greater than one or less than one. On the unit circle, y and x are either 1 or -1. When you divide using the given ratios, it still equals to 1 or -1. So that clarifies that the amplitude is always one or negative one because of this rule. But, for other trig functions it is different. There are boundaries to what values it must equal to because x and y can vary in different numbers. For example, cosecant, it is r/y. Your y CAN be 1 or -1, OR it can be any other number like 2 or 3. Same goes to the other trig functions. So there are no amplitudes. 

Reflection #1: Unit Q-Verifying Trig Identities

1. What does it mean to verify a trig identity?
To me, verifying an identity basically means to prove that an equation is true by showing that both sides are equal. I think of it as taking steps to simplify one side of the equation to make it equal to the other. Verifying a trig identity takes a gradual process of manipulation, alterations, and logical thinking! It may be difficult at first, but all you have to do is to make both sides equal and there are multiple steps so it is not as difficult as other math units.
2. What tips and tricks have you found helpful? 
I found that remembering the identities (ratio, reciprocal, Pythagorean) make verifying a trig identity much faster. Also, I learned that it's a little different than other units we have went through, because there are many ways to verify an identity and it's very ambiguous. Usually, we are usually used to using a formula and getting our answer right away, but with this we basically we have trial and error. Another tip is to not touch the other side! You simply cannot, no matter how tempting it is. 
3. Explain your thought process and steps you take in verifying a trig identity.
First, I look at the problem and I try to convert anything to a trig identity of sine or cosine.  Next, I usually look to see if I can take out a LCD (least common denominator). I factor if I need it. Then, I usually check if I can use any identities, especially Pythagorean identities. I also find that powering up can lead me to the Pythagorean identity. I sometimes multiply by conjugate if there is any fraction. If I get confused at all, I usually go to three steps back and review what I did. If I can't make sense of it, I usually start from scratch because I don't want to over complicate the equation and continue if I accidentally made an error.