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

1). Why is a "normal" tangent graph uphill but a "normal" cotangent graph is downhill?

The reason for this is that tangent and cotangent have different ratios that cause for the asymptotes to be in different places.

http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Bruce/instructunit/day_6.htm

•  The ratio for tangent in terms of sine and cosine is sine/cosine. As a result, tangent will have asymptotes whenever and wherever cosine is on the x- axis (when cosine is equal to zero).
• In order for tangent to follow its ASTC pattern  (positive, negative, positive, negative), the uphill shape shown above occurs in accordance with its asymptotes.

https://share.ehs.uen.org/node/19145

 

 

•  Cotangent, on the other hand has asymptotes wherever sine is equal to zero on the y-axis because the trig ratio is cosine/sine.
• The difference in the asymptote location causes for the differences in shape. Since cotangent and tangent follow the same ASTC pattern, they both are positive is quadrants I and III and negative in II and IV. In order for cotangent to follow this pattern, it must be downhill.

Used Websites: http://jwilson.coe.uga.edu/EMT668/EMAT6680.2001/Bruce/instructunit/day_6.htm
(tangent picture)
https://share.ehs.uen.org/node/19145 (cotangent picture)

BQ #5: Unit T Concepts 1-3: Trig Graphs and Asymptotes

1). Why are sine and cosine the only trig graphs that have asymptotes?

• As we can see from the above picture, sine and cosine have unique ratios when compared with all the other trig values. They both have a denominator of r. In the Unit Circle, r is always equal to 1. Now, the question is why don't sine and cosine have asymptotes whereas all the other ratios do?
• We get an asymptote whenever we divide by zero (in other words, whenever we have a denominator of zero.) This results in undefined answers, in which an asymptote accounts for.
• Sine and cosine have denominators of r which is always equal to one in this case. Therefore when using sine and cosine we will never be dividing by zero and never be getting asymptotes.
• The rest of the trig functions, however, have denominators of x and y. These denominators can be any number, and when that number is zero, then you will get undefined and therefore an asymptote.

BQ #3: Concepts 1-3: Graphing Trig Graphs

1). How do the graphs of sine and cosine relate to the others? (With an emphasis on asymptotes)

Well then, the very first thing you need to answer this is, what do ya know, the graphs. Who would've thought that, right? Anyhow, here goes nothing...

a). Tangent in relation to sine and cosine

• Here we see a graph of sine, cosine, and tangent (black) as well as some asymptotes (various shaded regions). Focus on the light green and orange regions. It is in these two regions that tangent is a full period, therefore we can observe the relationship it has according to its asymptotes through these regions. We can see that asymptotes occur wherever cosine (green) has a point on the x-axis (in other words a value of zero.)  This corresponding relationship can be explained in the trig ratio: tan = sin/cos. Since we know that we get asymptotes by dividing by zero, using this trig ratio we know that there will be an asymptote whenever cosine, the denominator, is zero.

b). Cotangent in relation to sine and cosine

• Here is the same graph with cotangent instead of tangent. We can apply the same logic to this one as we did above. Since cot = cos/sin, there will be an asymptote wherever the sine value is zero.

c). Secant in relation to cosine

• Shown here is cosine (middle and green) and secant (top/bottom and blue). Since secant is the reciprocal of cosine, it's relationship to the cosine graph is affected in various ways. The most obvious is the x-axis reflection of the peaks and valleys of the cosine graph. However, what we are looking for is how this relationship causes the asymptotes (black dotted lines). The reason asymptotes are there is once again because of a trig ratio, this time being that sec = 1/cos, and like before, since cosine is the denominator, any place where the cosine value is zero we will have an asymptote.

d). Cosecant in relation to sine

• This one is just like the secant and cosine graphs. The only difference is that the ratio applied is csc  = 1/sin and therefore wherever the sine value is equal to zero on the x- axis then there will be an asymptote.

BQ #2: Unit T Concept 1 Intro: Trig Graphs and the Unit Circle

1). How do Trig Graphs relate to the Unit Circle?

Trig graphs relate to the unit circle in that they follow the same pattern that is defined by ASTC in the Unit Circle in their periods, which is the length of each cycle in the graphs. As a result of this, the trig graphs are basically just "unraveled" versions of the unit circle.

Why do sine and cosine have periods of 2 pi and tangent and cosine have periods of just pi?

• This is can be explained through the Unit Circle and ASTC patterns that are repeated through the periods, as mentioned above. In order to fully understand this, the picture below shows this relationship.

• On the top is ASTC and the Unit Circle, and on the bottom are the trig graphs. Note how for sine and cosine it takes a full revolution of the unit circle in order to repeat the pattern of the signs. A full revolution is 2 pi. Since trig graphs follow the same pattern, it would take 2pi to repeat the +/- pattern and thus the period would be 2pi. However, take note of the tangent and cotangent pattern. This pattern only requires 1/2 a revolution, or just pi, to repeat itself and thus the period is just 1 pi for tangent and cotangent.

Why are Sine and Cosine graphs the only ones with amplitudes (in reference to the unit circle)?

• When we look at the different types of graphs, we notice that sine and cosine have infinite domain but restricted range. This calls need for amplitudes, which define the range of such graphs.
• But why does this occur? If we refer back to the trig ratios of the Unit Circle, we know that sine and cosine are the only two with denominators of r, which is always 1 on the Unit Circle. As a result, Sine and Cosine could be no bigger than 1, otherwise the ratio would cause for a point not on the unit circle resulting in no solution.  Thus, amplitudes are needed in trig graphs in order to ensure that the graph stays in this zone.

Click to see website

This illustrates the above mentioned information of the Unit Circle's limitations on sine and cosine. Amplitudes are needed to make sure the graph stays within such restriction.

Citations: http://htmartin.myweb.uga.edu/6190/functions3.html Unit Circle Trigonometry

Reflection #1: Unit Q: Verifying Trig Identities

1). To verify an identity means to prove that it is true. For example, an identity might be one equation or term that is equal to another and you have to prove that to be true using your knowledge of identities and your knowledge of simplification processes.

2). I have found that in verifying identities there are always the same strategies being repeated. These strategies include finding a GCF or a LCD, multiplying by the conjugate, separating fractions, converting into sine and cosine, using the zero product property, etc. Of these, I have found that converting into sine and cosine and separating fractions (only those with a monomial denominator) are two of the most recurrent and helpful strategies at approaching a problem. I also like to look at the answer when verifying to see if I can find any hints as to what to do to get there. Lastly, I also always look for things that might cancel each other out, since they can be very helpful.

3). I always ask myself when doing these problems "what can I do to change the problem in order to be able to use identities to get the answer?". Usually I look for anything that might hint toward a specific identity. For example, if I see cosine squared  and sine squared I know I am going to be using a Pythagorean identity. Sometimes the problem is more complicated and you might have to alter it a bit before any identity jumps out in front of your face. A common example of this would be if you have to factor the problem first in order to get something that you can use.

SP #7: Unit Q Concept 2: Finding all trig functions using idenitites

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 This is our problem. We are given that cotangent is 7/8 and sine is negative. Also we know that the problem lies in the third quadrant since tangent is positive and cosine is negative according to ASTC. We use what we know about identities to solve for the rest of the trig functions. We will then check our answers by solving with the unit circle, which we learned in Unit N or by using SOH CAH TOA from Unit O.

 One of the easiest things to do is find tangent which is the inverse of cotangent. Knowing this, when we use the reciprocal identities we find that tangent is equal to 8/7.

 Now that we know tangent we can use a Pythagorean identity to find secant. Secant ends up being equal to radical113 divided by 7.

 Cosine is the inverse of secant, which we just found, so we are able to use a reciprocal identity using secant to find cosine. Cosine will be 7 rad 113/ 113.

 First here we find sine using a Pythagorean identity and we can do this since we have found the value of cosine. We then find cosecant by doing the inverse of sine, also known as using a reciprocal identity.

 We have now found all the trig functions using identities. Note that all the answers except tangent and cotangent are negative because the problem lies in quadrant III.

Here is the same problem done using SOH CAH TOA. Note that the answers are identical with the answers found using identities.

1).  It is important to know several things in doing this problem. One must pay attention to what identity is being used to solve for each function. There are different ways to find each function with different identities but one must make sure that there is only one unknown trig function when plugging into an identity. Also, one must make sure to get their signs correct by understanding how we found what quadrant the answer was in and what effect that quadrant has on the answers.

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

Inquiry Activity Summary:

 1). An identity is a proven fact or formula that is always true. Pythagorean Identities are proven to be true and are one type of identities that we are studying in this concept. They can be easily derived through our knowledge of the Pythagorean theorem and the Unit Circle. The following will show precisely how I have done just that.



 

 

 

• So basically what I did was I wrote out the Pythagorean theorem and then made it equal to 1 by dividing everything evenly by r squared. I then used what I knew of the trig functions, specifically the ratios of sine and cosine to rewrite the equation. The end result was the first Pythagorean identity.
• The next thing I did was I tested the truth of the equation. I used what I knew of the Unit Circle to get a point (in this case the point for 45 degrees) and plugged in the cosine and sine values of that point into the equation. The equation met expectations, meaning that the equation is indeed an identity.

 

2). The other two main identities of the Pythagorean Identities can be derived by



simply by dividing by either cosine squared or sine squared from the first identity.

• After dividing by cosine, the cosines will cancel into a 1 and using the ratio and reciprocal identities we find that we are left with tangent and secant. The result is the second Pythagorean identity.
• After dividing by sine we find that like the other one, something cancels into a one and the trig functions change (this time into cotangent and cosecant). The result this time is the third main Pythagorean identity.

Inquiry Activity Reflection:

1). The connections I see between Units N, O, P, and Q so far are that the identities are related to the Unit Circle (Unit N) in that we use an r value of 1 for the Pythagorean theorem and we use the sine and cosine of a point on the unit circle to test the equation and that we use the relationships of the various trig functions to do the derivations. We also use the Pythagorean in the other derivations we have done for the various other units.

 

2). If I had to describe trigonometry in three words they would be: integrated, complex, and understandable.