BQ#4 – Unit T Concept 3

Why is a “normal” tangent graph uphill, but a “normal” Cotangent graph downhill? Use unit circle ratios to explain.
Asymptotes play such a large part in the reason why "normal" tangent graphs go up and why "normal" cotangent graphs go down.
Tangent, as discussed before, has the ratio of y/x or sine/cosine. With this knowledge, we know that whenever our x-value/cosine is 0, we will have an asymptote. Well, if we think about it, at pi/2 (0,1) and 3pi/2 (0,-1), our x-value is 0. These are where are asymptotes lie. Now, thinking back to our unit circle, we know that from pi/2 to pi, we go into Quadrant II, where only sine is positive--in other words, tangent is negative. So, in our first half of the space between our two asymptotes, our values are negative. Moving on, we know that pi to 3pi/2 in the unit circle is Quadrant III where tangent is positive. So, our other half of the space between our two asymptotes is going to be positive. Looking at the entire picture, we would see that our tangent graph begins at the bottom and goes up, making it uphill!
Cotangent has the same idea. Its ratio is x/y or cosine/sine. We want sine/our y-value to be 0 in order to find our asymptotes. So, the only places in the unit circle where we have 0 as our y-value are at 0 radians (1,0) and pi (-1,0). Therefore, our asymptotes are at 0 and pi. Again--same idea. From 0 to pi/2, we have Quadrant I where all trig functions are positive; cotangent is positive. However, from pi/2 to pi, we have only sine being positive so cotangent is negative. Guess what? That is what we are going to be seeing on our cotangent graph. Our first half of the space between the asymptotes is basically Quadrant I, so our graph will have positive values. However, our second half of the period is in Quadrant II where we established the fact that cotangent is negative. Our graph will go down. So, looking at the entire graph as a whole, we see that our graph starts from top to bottom, going downhill.

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.
Sine and cosine are found in the ratio identities of the rest of the four trig functions; they affect each one of those trig functions because of this.
Tangent
With tangent, we know that tangent is equal to sine over cosine. Now, just by knowing the values of sine and cosine, we can assume what tangent will be on the graph. Look at the visual below to better your understanding.

(https://www.desmos.com/calculator/cwdr1eyszr)

        If you look at the red highlighted area, that is 0 to pi/2, so basically the first quadrant. Even without looking at this graph, we know that sine and cosine are positive, so tangent must be positive as well (positive divided by a positive is a positive). And you can see this translate on the graph. If you look at the green cosine values and the red sine values, you can tell they are above the x-axis, meaning they are positive. And, we determined that since those to trig functions are positive, tangent must be positive, which it is (the orange line). You can see the same thing going on in the next parts of the graph. In the "Quadrant II", on the graph, you can tell that cosine is negative (as the green values fall below the x-axis) and sine is positive (as the red values are above the x-axis). And because of our ratio identity of tan(x)=sin(x)/cos(x), we know that a positive divided by a negative makes a negative. That is why the tangent values fall below the x-axis this time. The 3rd quadrant, again, look at our sine and cosine values: they are both negative. So, a negative divided by a negative is a positive: tangent values are positive. Finally, you should be understanding the relationship that tangent, cosine, and sine have. The last part of the period, cosine is positive (values are above the x-axis) and sine is negative (values below the x-axis); tangent is negative. 

         Now, that you know why the tangent graph looks as it does (through the understanding of its relationship with sine and cosine), we need to deal with another part of these graphs: the asymptotes. The asymptotes are not just there because we say so. There is a reason behind it. Remember when we talked about how tan(x)=sin(x)/cos(x)? We are using this identity again. We know that asymptotes are created when we have an undefined value. We get undefined values when we divide by 0. So, knowing that... we look to our ratio identity and see that cosine must equal 0 in order to get an undefined value (because we are dividing by cosine). How do we get cosine to equal 0? Well, we know that cosine is essentially the x-values of certain points in our unit circle. The only parts of the unit circle that have 0 as an x-value are pi/2 (0,1)and 3pi/2 (0, -1). Therefore, these marks on the graph are where we have asymptotes! Look at the graph above and see where it seems like the tangent graph discontinues; do you notice that pi/2 and 3pi/2 are where we don't really see the tangent graph go? That is because of the asymptotes! 

Cotangent

Cotangent has sine and cosine in its ratio identity as well. Cot(x)=cos(x)/sin(x). 

(https://www.desmos.com/calculator/cwdr1eyszr)

Cotangent has the same thing going on as we saw in tangent. Remember to look at the values of the sine and cosine graphs and relate them to cotangent. For example, see how sine and cosine values are positive, so we have cotangent as positive. However, in the green section, we see that sine (the red graph) has positive values and cosine (green graph) has negative values; we can deduce that cotangent is negative because a negative divided by a positive is a negative. 

Because cotangent has a different ratio than tangent, we are going to be having some different ratios. The ratio identity of cotangent is cot(x)=cos(x)/sin(x). Again, asymptotes are undefined values, meaning they are divided by 0. So, because sine is our denominator in this ratio, sine must be equal to 0 in order to find asymptotes. Where on our unit circle do we have sine=0? Sine is y/r, basically the y values of the unit circle. So, we know the only parts of the unit circle where the y values are 0 are 0 (1,0) and pi (-1,0). Knowing this, we know that our asymptotes are at 0 radians and pi. Look at the graph above and check to make sure. Notice how at 0 radians, a new period begins, as if we lifted our pencil to create a new part of the graph. The same is seen at the pi mark. 

Secant

Secant is a little different from tangent and cotangent. Its ratio identity is 1/cos(x).

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

We have the same idea though, as we saw in the previous graphs. If cosine is negative, then secant values must be negative as well. If cosine is positive, the secant graph must be positive as well. You can see this in the graph. The cosine values (the black graph) are positive in the red shaded part of the graph; and look at that, the secant graph (the red graph) is positive in that section as well. You can see the trend in the following sections/quadrants as well. It is good to remember that the sine values in here do not have any effect on the secant graph because it is not part of its ratio (*disregard the orange sine graph in the picture above*). 

The asymptotes of secant are the same as we saw in tangent. Tangent had cosine as the denominator as well. So, we know that at pi/2 (0,1) and 3pi/2 (0,-1) there are going to be asymptotes for this secant graph. 

Cosecant

Cosecant has the same idea as secant just reversed, like we saw with cotangent and tangent. Cosecant's ratio identity is csc(x)=1/sin(x). 

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

Again, we remember that if sine is positive on the graph, then cosecant must be positive as well, because 1 divided by a positive is always a positive--and vice versa. You can see in the red shaded part, the sine graph (the orange line) is positive and the cosecant graph (the red line) is positive as well. It is good to remember that cosine has no effect on cosecant because it is not found in the ratio (*therefore, disregard the cosine graph in the picture above*). 

Cosecant has its asymptotes wherever sine is 0 (because we are dividing by sine in this ratio). Looking back to our cotangent graph, we remember that sine was 0 at 0 radians and pi (because of the coordinates of (1,0) and (-1,0)). So, we will have cosecant's asymptotes at 0 radians and pi. That is why, if you look at the picture above, you see that the graph seems to cut out at 0 radians and pi. 

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 their ratios are y/r and x/r, meaning that the radius will ALWAYS be the denominator. You will never have a 0 as your denominator. Because of this, we will never encounter an undefined answer, meaning we will not encounter any asymptotes (since asymptotes = undefined, basically).

However, that is a different case for cosecant, secant, cotangent, and tangent. Cosecant has a ratio of r/y. The "y" value can be 0 in certain cases: if we are at (1,0) or (-1,0). Same for secant; secant has a ratio of r/x and x can be 0 in some cases: (0,1) and (0, -1). Cotangent and tangent have the ratios of x/y and y/x so, again, of course you will have some y values and x values that equal to 0.

Also, a good thing to nice is that cotangent and and cosecant have the same denominator in their ratios: y. Therefore, they will have the same asymptotes. Tangent and secant will have the same asymptotes as well because their denominators in their ratios are x.

BQ#2 – Unit T Concept Intro

How do the trig graphs relate to the unit circle?
Trig graphs, believe it or not, are essentially the unit circle unwrapped (at least in one period). BAM. Let's think about what we already know about the unit circle. We are going to use sine as an example, first. In the case of sine, the 1st and 2nd quadrants are positive whereas the 3rd and 4th quadrants are negative. Now, the 1st quadrant goes from 0 radians to pi/2. The 2nd quadrant goes from pi/2 and pi. So from this information alone, we can deduce that from 0 to pi, our values are going to be POSITIVE. The 3rd quadrant goes from pi to 3pi/2; the 4th goes from 3pi/2 to 2pi. Because sine is negative in these quadrants on the unit circle, in the trig graphs, from pi to 2pi, our values are going to be negative. Thus, the whole entire graph will go up and then down, because it is corresponding with the values we know from the unit circle.

 

As you see, the same can be seen for other trig functions. For cosine, we have the pattern being positive, negative, negative, postive. Therefore, on the graph, you will see the period beginning with positive values, but dipping at pi/2 and becoming negative. At pi, it remains negative, but when you hit the mark of 3pi/2, our values will be positive, as it should.

 

Tangent and cotangent are similar however there is a special situation with them (which I will go into further detail in the next section). Anyway, our graph for tangent/cotangent will begin with positive values, then negative values, then positive, and negative. However, if you notice, we have a pattern already seen in the first two quadrants, that is then repeated in the next two quadrants--which leads us into the next question!

 

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 the pattern finishes at 2pi and does not repeat until we begin another "revolution" around the unit circle. The pattern is +,+,-,- for sine. The pattern is +, -, -, +. You see, you need all quadrants in order to explain the FULL pattern. You need to go around the unit circle in ONE FULL revolution. So, your period will go to 2pi.

The period for tangent and cotangent is only pi. Why? Look at the picture above. Do you notice that the pattern is +, -, +, -? The pattern repeats itself twice in one revolution of the unit circle. Why do we want to repeat that? So, we only really need the first quadrant in order to fully explain the pattern. That is why tangent and cotangent periods only go to pi.

 

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 why is it that we have amplitudes of one for sine and cosine? Think about it. Sine has the ratio of y/r. Cosine has the ratio of x/r. R can only equal 1, since the radius of a unit circle has to be 1. So, the values x and y can only go up to 1 themselves. So if you can only divide 1 by 1 to get the largest number... your largest value can only be 1/ -1. That is why sine and cosine cannot equal something greater than 1 or less than -1. However, think about the other trig functions and their ratios. Tangent is y/x. You are not restricted to values between -1 and 1 because you are dividing by different numbers that vary from being just 1 and -1. Same is for cotangent but backwards. Also, for trig functions like cosecant and secant, you have r/y and r/x. You can divide your "r" (1) by a small number like 1/2 and end up with 2. That is why those other funcitons do not have amplitudes.

Reflection #1: Unit Q: Verifying Trig Identities

1. When a problem asks you to "verify a trig identity", it basically means to work out one side of the equation, simplifying it to the point to where it equals the other side. It is almost as if the problem gave you the starting and ending points and you are just finding the steps to get from one to the other.

2. First of all, make sure you memorize all the identities. They're honestly not that difficult to remember after dealing with them SO MUCH, and it is a nice thing to know that this equals this at the top of your head. However, sometimes just knowing these identities, does not do anyone any good. I find changing everything in terms of cosine and sine makes everything a lot easier. What I like to say is "When in doubt, do sine and cosine." and it's pretty much true. Also, u substitution was such a blessing on my mind. It basically allows us to replace a trig function with "u" when we have a quadratic on our hand. Trust me, you do not want to be "x-boxing" or factoring or FOILing (sinx +1)^2. I don't know if these can actually be called "tricks" that helped me get through this unit, but I know these reminders saved my life on the test. For verifying, it is essential for you to know that touching the "right side" (the side that you're usually trying to work towards), is a BIG NO NO. Another thing I found helpful was knowing that whenever you square anything, you are going to end up with extraneous solutions; as a result, you will need to make sure you actually plug your answers into the original equation to ensure the "correctness" (is that a word...?) of the answer.

3. Whenever I see a problem that asks to verify, I get excited because they practically told you what you're supposed to end up with! Now you just have to find your way to the end. My general thought process is, "Are there any trig functions that can convert to sine and cosine?". If I see cotangent or cosecant, and so on (you get the point), I would usually replace them with sine and cosine. However, sometimes that does not work for me, so my next thought is "Are there any trig functions that are squared?" Usually, if something is squared, I would replace that with a Pythgagorean Identity, and then go from there. If I see that I have a binomial as my denominator, I will multiply the entire fraction by the conjugate. If I see two/ multiple fractions being added together, I will make sure the two have the same denominator, and if they don't (which is 90% sadly), then I would have the lowest common denominator (LCD) between the two, multiply each fraction with what is needed to achieve that LCD, and then combine them so I have one large fraction.  Again, I really only focus on the left because we don't really need to do anything to the right. This is just really what I think in general, and my thinking will change depending on the types of problems I am given. Ultimately, the best piece of advice I could give to you when dealing with verifying is just to go through it slowly and rationally. Don't feel the need to rush and you'll be able to get through these like nothing.