Sunday, April 20, 2014

BQ #4: Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?

Tangent






















Cotangent




                                                                                                                                                                                                                              








Let's go back to the unit circle - what appears to be the solid foundation of pre-calculus. 
In the unit circle tangent and cotangent were related by its ratios. Cotangent's ratio was the reciprocal of tangent's ratio. Tangent's ratio is y/x meaning cotangent's ratio would be x/y. Now in this unit we would read that as tangent's ratio being sine/cosine (according to Unit Q's identities) and cotangent's ratio would be cosine/sine
Now what these two graphs have (as well as secant and cosecant) are asymptotes. Remember that this occurs when dividing by 0 leading to undefined. Tangent's asymptotes would occur when cosine equals 0 which would be at pi/2 and 3pi/2. Cotangent's asymptotes would occur when sine equals 0, this would be at 0, pi, and 2pi. (Reflect back on the unit circle, think of these values.) 
Because their asymptotes are placed in different areas, this is going to affect the direction in which they go. Also remember that these graphs WILL NEVER TOUCH the asymptotes, EVER. They will get VERY close but NEVER TOUCH. 
Not only do the asymptotes affect its direction, but look at the different colored areas in the images. Red is the first quadrant of the unit circle, where all trig functions are positive. Green is the second quadrant, where tangent and cotangent will be negative (as well as cosine and secant, but not sine or cosecant.) Orange is the third quadrant, where tangent and cotangent will be positive (the rest of the trig functions will be negative.) Blue is the fourth quadrant, where tangent and cotangent will be negative (as well as sine and cosecant but not cosine or secant.) 

Re-using the images from BQ #3, all thanks again to the wonderful Mrs. Kirch. Her amazing help with these graphs can be found here. 

Saturday, April 19, 2014

BQ #3: How do the graphs of sine and cosine relate to each of the others?

Sine and Cosine

Here we have two trig functions: sine and cosine. Because sine and cosine are always divided by r which is one they will never be divided by 0 hence never being undefined hence never needing an asymptote. There are five possible asymptotes which are depicted as dotted lines at the points of (0,0), (pi/2, 0), (pi, 0), (3pi/2, 0), and (2pi, 0). These points are were asymptotes can be found depending on the trig function selected. The four different colored sections represents one of the four quadrants from ASTC. The red is the first quadrant, green is the second, orange is third, and blue is the fourth. 

Remember that an asymptote results when a ratio is divided by zero, becoming an undefined ratio. 

Tangent

Tangent's ratio is y/x, meaning that there is a possiblity of an asymptote when x (cosine equals 0). Because of this we know cosine equals 0 at 90* and 270* so pi/2 and 3pi/2, this is where are asymptotes will be located. Now let's look at the first quadrant, remember that in the first quadrant all is positive so it goes in an uphill direction and will NEVER touch the asymptote of pi/2, it simply gets really, really, really close to it. In the second quadrant, both sine and cosine are heading in a downhill direction and in the second quadrant tangent is not positive so it heads downwards but in the third quadrant it is positive. The graph can continue in these two quadrants because there is no asymptote dividing them. And in the fourth quadrant, tangent is not negative hence it's downhill direction. 


Cotangent
For cotangent our ratio is x/y, meaning we will have our asymptotes where y =0 (sine) and those locations would be at 0*, 180*, and 360*. On our graph these would be the values of  0, pi/2, 2pi. Because both sine and cosine are positive in the first quadrant as everything is, cotangent is positive as well. Yet in the second quadrant, sine is positive and cosine is negative leading contangent to continue in the negative direction crossing the x-axis when cosine does as well. The first and second quadrants already contain one period of cotangent. A similar process continues off from the asymptote of pi in the third and fourth quadrants. Because sine and cosine are negative in third quadrant, cotangent will be positive and because sine is negative and cosine is positive in the fourth quadrant, cotangent will be negative. 


Secant

Remember that secant is the reciprocal of cosine's ratio which will be r/x meaning that there will be asymptotes where cosine is equal to 0, similar to tangent's asymptotes. A similar pattern also follows here.
In the first quadrant, both sine and cosine are positive and so will secant. But however in the second quadrant, sine and cosine are both negative, as will secant and will continue to be negative because although cosine is positive in the third quadrant, sine is negative and a negative and positive will result in a negative. In the fourth quadrant, both sine and cosine are moving in an uphill direction and secant will also be positive. Once again notice how none of secant's graph is touching the asymptotes and how they develop at the mountains and valleys of cosine's graph.

Cosecant 

Cosecant is the inverse of sine meaning its ratio will be r/y. This being said, cosecant will have asymptotes wherever sine equals 0, also similar to cotangent (oooh connections!) In the first quadrant secant will remain positive because all functions are positive in the first quadrant. Yet although cosine is negative and sine is positive in the second quadrant, cosecant will still be positive because it is positive in the sine quadrant of the unit circle. The graph continues into the third and fourth quadrant even after having gone through its period and it's direction is enforced by the unit circle's positive or negative values of the cosecant function. However more importantly because cosecant is the inverse of sine, it relies on the sine graph to be drawn because once again notice, like secant, it is drawn on the mountains and valleys of its corresponding reciprocal of the sine function. 


All images made available thanks to the amazing and wonderful Mrs. Kirch on Desmos, you can view and animate as well here.










Tuesday, April 15, 2014

BQ #5: Unit T: Concepts 1-3: Why do sine and cosine NOT have asymptotes, but the other four trig graphs do?

Image found here.


Image found here. (thank you Google!)

Sine and cosine do not have asymptotes because they are always divided by 1 (by "r")

Yet once we move away from being divided by "r" we start reaching asymptote area.
If you reference to the photos you will notice undefined under tangent. Do not forget about their reciprocal ratios of cosecant (r/y), secant (r/x) and tangent (x/y).
Remember that we get asymptotes when our circle ratios equal undefined which results when you divide by 0. 
For sine and cosine you will never be dividing by 0 as you may for all other four circle ratios. But because you'll never be dividing by 0, you will never reach undefined, so you'll never have asymptotes! 

BQ #2: Unit T: Concept Intro: How do trig graphs relate to the Unit Circle?

Sine and Cosine In these images you see how the Unit Circle translates and unravels itself into a sine curve graph. 

Period: The period for sine and cosine is 2pi because it takes four quadrants (ASTC) to repeat the pattern. Sine's pattern according to ASTC from the unit circle is + + - -

 While cosine's pattern is + - - +
It merely takes all of the unit circle (which is 2pi at 360* to complete sine and cosine's patterns.
Graphs are merely snapshot of the graph, these graphs are infinite as circles are - no ends, no beginning but for the sake of this class we will only be graphing one single period.


 Amplitude: Remember that sine equals y/r and cosine equals x/r and that r=1. In the unit circle, the values cannot be bigger or smaller than 1, thus x and y always equal 1. 1 divided by 1 equals *drumroll please* ONE!!!
Meanwhile all the other trig functions have asymptotes because they do not have "r" such as tangent and cotangent (y/x and x/y respectively.)



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Cosine





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Tangent/Cotangent

Meanwhile with tangent and cotangent.....
Period: Tangent and cotangent have a period of just pi. According to ASTC from the Unit Circle, the pattern this trig function has is + - + -. Meaning the pattern is completed in the first two quadrants which is 180* aka pi aka half of the circle.
Just remember again to keep in mind that a period is one time through their cycle, their pattern. 
 













Images found here.