In the Unit Circle, tangent and cotangent are both positive in the first quadrant and negative in the second. Then it is positive in the third and then negative in the fourth quadrant. The identity for tangent is tan(x)=sin(x)/cos(x) and the identity for cotangent is cot(x)= cos(x)/sin(x). This causes the asmyptotes to be place in different positions. The asymptotes for tangent are in different places than the asmyptotes for cotangent and this is what makes the trig graph go uphill or downhill.
The reason why tangent is going uphill and it is normal is because the first asymptote is place at pi/2 which is the end of the first quadrant. In the first quadrant the function needs to be going up, in order to go on to the next part of the function it needs to start down without touching the asymptote. That is why it starts over and starts down then goes up this is why it is going uphill.
Cotangent is different. The first asymptote is mark at pi. This means the function can start up in quadrant one and progress down to quadrant two. There is no need for a new function to start because there is no asymptote separating those two quadrants. This is why it is normal for the function to be going down hill.
Sunday, April 20, 2014
BQ#3: Unit T: Concepts 1-3
Sine and cosine are a part of every graph. When we go back to our identities we know that
a. tan(x)=sin(x)/cos(x)
b. cot(x)=cos(x)/sin(x)
c. sec(x)=1/cos(x)
d. csc(x)=1/sin(x)
Asymptotes from sine and cosine cause these graphs to differ. Sine and cosine are the only functions without a x or y in its denominator. All other trig functions do. Whenever sine (y) is 0 there will be an asymptote and that is where cotangent and cosecant have their asmyptote. The same goes for cosine (x), whenever it is 0 there will be an asymptote. This is what causes the trig graphs to change and differ.
a. tan(x)=sin(x)/cos(x)
b. cot(x)=cos(x)/sin(x)
c. sec(x)=1/cos(x)
d. csc(x)=1/sin(x)
Asymptotes from sine and cosine cause these graphs to differ. Sine and cosine are the only functions without a x or y in its denominator. All other trig functions do. Whenever sine (y) is 0 there will be an asymptote and that is where cotangent and cosecant have their asmyptote. The same goes for cosine (x), whenever it is 0 there will be an asymptote. This is what causes the trig graphs to change and differ.
BQ#5: Unit T Concepts 1-3
Whenever you have 0 in your denominator it will be undefined. You can only have an asymptote if you have an undefined number. All the trig functions have a ratio:
Sine Cosine Cosecant Secant Tangent Cotangent
y/r x/r r/y r/x y/x x/y
When we are talking about the Unit Circle "r" matter what it equal to 1. When we look at the ratios, sine and cosine are the only trig functions where r is the denominator. This means no matter what the numerator is, the function will never be undefined because the denominator will always be 1. All the other functions have the possibility of being undefined because x or y can equal to 1 or 0 and that all depends on what point on the graph you are on. If you are on 0 or 180 or 360 degrees, cosecant and cotangent will be 0 because the ordered pairs on that point are (1,0) and 0 is y. If you are on 90 or 270 degrees, secant and tangent will be 0 because the ordered pairs are (0,1) and x is 0.
When we are talking about the Unit Circle "r" matter what it equal to 1. When we look at the ratios, sine and cosine are the only trig functions where r is the denominator. This means no matter what the numerator is, the function will never be undefined because the denominator will always be 1. All the other functions have the possibility of being undefined because x or y can equal to 1 or 0 and that all depends on what point on the graph you are on. If you are on 0 or 180 or 360 degrees, cosecant and cotangent will be 0 because the ordered pairs on that point are (1,0) and 0 is y. If you are on 90 or 270 degrees, secant and tangent will be 0 because the ordered pairs are (0,1) and x is 0.
BQ#2: Unit T Intro
1. The Unit Circle is a trig graph unraveled. It follows the same patterens as it does on the Unit Circle. When you have sine on the Unit Circle, it is going to be + + - -, on the trig graph it is the same thing it is up up then goes down down and that ends that period of the graph and on the Unit Circle it marks then end of a whole circle.
The period of sine and consine is 2pi because then you look at the negatives and positive from the Unit Circle, it takes one whole circle for the pattern of positives and negatives to start again. When you go around the Unit Circle completely that is 2 pi, that is why the period is 2pi. The reason why tangent and cotangent is pi is because it takes half the circle for the pattern to start over.
Sine and cosine are the only ones that have an amplitude, they are also the only two graphs that do not have that do not have asymptotes. All the other graphs have asymptotes because they certrain points where the graph is undefined and does not touch those points. With sine and cosine they graph will touch any points as long as they are within the amplitude. With all the other trig functions they have asymptotes. Amplitude and asymptotes are restrictions for each graph.
The period of sine and consine is 2pi because then you look at the negatives and positive from the Unit Circle, it takes one whole circle for the pattern of positives and negatives to start again. When you go around the Unit Circle completely that is 2 pi, that is why the period is 2pi. The reason why tangent and cotangent is pi is because it takes half the circle for the pattern to start over.
Sine and cosine are the only ones that have an amplitude, they are also the only two graphs that do not have that do not have asymptotes. All the other graphs have asymptotes because they certrain points where the graph is undefined and does not touch those points. With sine and cosine they graph will touch any points as long as they are within the amplitude. With all the other trig functions they have asymptotes. Amplitude and asymptotes are restrictions for each graph.
Wednesday, April 2, 2014
Reflection #1: Unit Q-Verifying Trig Identities
1. To verify a trig function is to show that the function given on the left side matches up the function given on the right side by simplify, substituting, and multiplying. You never touch the right side. The point is to get both sides to equal each other.
2. What has really helped me solve these functions was to memorizes all the identities. At first I thought it was pointless but when you are in the process of verifying or simplifying a trig function its a hassle to constantly having look back and forth. MEMORIZE! MEMORIZE! MEMORIZE!
3. The first thing I always do is try to change everything to sine and cosine. It makes everything a lot easier when it is in sine and cosine. Then I look for GCF. From there I look to cancel out anything and or combine like terms. From there I see if i can substitute an identity again and I'm usually done by this point.
2. What has really helped me solve these functions was to memorizes all the identities. At first I thought it was pointless but when you are in the process of verifying or simplifying a trig function its a hassle to constantly having look back and forth. MEMORIZE! MEMORIZE! MEMORIZE!
3. The first thing I always do is try to change everything to sine and cosine. It makes everything a lot easier when it is in sine and cosine. Then I look for GCF. From there I look to cancel out anything and or combine like terms. From there I see if i can substitute an identity again and I'm usually done by this point.
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