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In the picture above we see a sine graph (the red one) and a cosine graph (the green one). There is also four regions shaded. The red is the first quadrant according to the unit circle, the green the second quadrant, the orange the third quadrant, and the blue the fourth quadrant. Both graphs follow the unit circle of positive where they need to be and negative where they need to be. Remember that sine goes positive positive then negative negative and takes 2pi to repeat and cosine is positive negative negative positive and also takes 2pi to restart. They abide by the rules of being greater than negative one and less than one. It is the basic graph.
Asymptotes
Now when we look at the graph we see that there are five different dotted lines and those are all places that asymptotes can exist depending on the trig function that you are using. The first, third, and fifth asymptotes exist only when we have a cosecant and cotangent graph because their ratios have y in the bottom and when y is zero then the ratio becomes undefined and we have a whole. The second and fourth are asymptotes that are seen when we have tangent and secant graphs because the ratio places x in the bottom and those are the spots where x is zero making it an undefined value.
Tangent
Since the asymptotes exist at pi/2 and 3pi/2 the graph needs to avoid these spots on the graph. These asymptotes are also the spots where cosine cross the x-axis at zero. This is because the tangent ratio is sine/cosine and on the unit circle cosine is zero at those two locations. The same apply to the graph.
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In this photo we have the first quadrant of the tangent graph. It needs to stay positive for the first quadrant and avoid the asymptote so it goes uphill and does not touch the line of pi/2.
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Here we can see that the second part of the graph is drawn. It needs to be negative and it also needs to avoid the asymptote of pi/2. In order to make this possible it goes below the x-axis. This is fair to the ratios on the unit circle because one is negative and one is positive.
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The next part of the graph is the piece from the third quadrant. This continues from the second quadrant and pulls into the positive because that is what tangent ratios appear as on the unit circle. It can continue from the previous portion of the graph because there is no asymptote at pi because the unit circle ratios here are 1/1 and that is a number that exists and can be passed through. It does however have to avoid the asymptote that is created by the sine being 1 and the cosine being 0, creating an undefined term.
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This is the final part of one period of the graph. This means that the graph can start over again. The negative has been hit again. The fourth quadrant on the graph is a negative because it is a negative number over a positive number under. This means that it goes under the x-axis and turns up at the x-axis because no asymptote exists at 2pi for tangent graphs. It does however, have to make sure to avoid the asymptote at 3pi/2 (the same reason as the last portion of the graph). That is really all to the tangent graph. It will go like this on and on.
Cotangent
For the cotangent graphs there are three asymptotes within one revolution of the unit circle. These exist because they are the three locations on the unit circle that the ratio has a zero in the bottom (as the sine value). When you divide something by zero we have something known as an asymptote. The graph must stay in the positive and negative regions-according to the unit circle-and avoid the asymptotes. These asymptotes exist wherever sine touches the x-axis at zero because the asymptotes are created by sine being zero and making it undefined.
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The first part of the graph lies in the upper portion of the graph because on the unit circle it is positive due to both sine and cosine being positive values. It starts at the top, avoiding contact with the zero asymptote (0 degrees) and then going negative after the crossing of the x-axis. It crosses the axis at the same point that the cosine touches the x axis.
For the second part of the graph it continues from the first part and goes negative because cosine is negative and sine is positive. This portion also has to avoid the asymptote at pi (created because sine has a zero value making it undefined). This creates one whole period of the cotangent graph.
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The third part of the graph is again above the x-axis because sine and cosine are both negative making it a positive. It must avoid the asymptote on its left so it gets close, but just like the previous portion it cannot touch it. Then it heads down and passes through the x-axis at the same point that cosine passes through the x-axis.
The fourth part of the graph is continued from the previous portion. It is negative because sine is negative and cosine is positive. Then it goes only as far as the asymptote that is created by the sine being zero on the unit circle It cannot touch that one. Now we have the full two periods of the cotangent graph.
Cosecant
This is the inverse function of sine. It means that it will have asymptotes that are the same as cotangent because cosecant, like cotangent, has the sine on the bottom and will have the zeros occur in the same spots. It will follow the same pattern as the sine graph of positive positive and negative negative to have a 2pi period, but it does have its differences. It needs the sine graph though so that the real graph can be created.
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In this picture the first quadrant is shown with the first part of the cosecant graph. It is avoid the asymptote because that is an undefined spot as we discussed in cotangent graphs. It is starting from the maximum of the sine graph and only displaying one half of it. It is there because the ratios on the unit circle dictate that it is positive and that it must avoid the zero asymptote. This is one portion of the graph though. Between the asymptotes there are two halves. the other half lies in the second quadrant.
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In the second quadrant we see a full parabola. The graph remains positive still because the unit circle dictates that it will remain positive. The second portion of the parabola starts at the maximum of the sine graph as well. Now we have a positive parabola that originates off of sine's maximum and avoids the two quadrant angles that create an undefined value (the asymptotes).
In the third quadrant we see the cosecant graph turn negative just like the sine graph (because they are the same values just in different positions in the ratio) and it starts a negative parabola. It touches at the minimum of the sine graph and turns downwards. It avoids the asymptote on the left because it is still an undefined value dictated by the unit circle values of y. This is again only a piece of it.
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The fourth quadrant is also negative because of the unit circle values that are present in the ratio. It also starts from the minimum of the sine graph and turns right down and avoids the 2pi asymptote that is created by the y being a negative number. This forms the last part of the parabola. Now we have our full period for a cosecant graph.
Secant
When we are looking at our secant graphs they will have the same asymptotes as the tangent graph because both have x on the bottom and that means that they will have undefined ratios in the exact same spots. This graph will be avoiding those spots and will follow the cosine graph in the positive and negative areas of the quadrants on the graph (for that is what is dictated by the unit circle). It is not too difficult to understand, but you will need your cosine graph in order to create the secant graph.
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https://www.desmos.com/calculator/hjts26gwst)
In the picture we see that the first part of the graph is positive like the cosine graph and that is because of the values that are given in the unit circle. Then we see that it is coming off the maximum of the cosine graph. We notice that it is avoiding the asymptote because that is a zero on the bottom of the ratio and that is not allowed. This is only part of the parabola that it forms. The other one is in the opposite revolution of the circle so it is not shown there, but it is there and it avoids the asymptote on that side of it. It is all pretty simple.
In the second photo we see that the graph avoids the asymptote that is not allowed to touch since that is a zero in the denominator (cosine). Then we notice that it is coming off of the minimum of the cosine graph because they contain the same x values since it is the unit circle that we base the parent graph off of. It is not too difficult to understand. It will have another half to make the complete parabola and that lies in the third quadrant.
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In the third quadrant (which is also negative due to the ratio values that are present) it forms the other half of the parabola. It stems off the minimum of the cosine graph. It curves down since it is negative and goes all the way to almost the next angle (on the unit circle) but avoids it for it is a zero value making it undefined and an asymptote. This gives it the parabola shape unlike the original graph.
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In the fourth quadrant we see that it is positive because the unit circle ratios make it positive. It does need to avoid the asymptote on its left though because that is an undefined ratio so it does not exist. It curves and sprouts from the maximum of the cosine graph and starts the parabola shape. It makes it the parabola shape that it needs to have. This is a whole period even though it does not look like it. It is.
Overall
In the end we see that the asymptotes that the rest of these graphs have are the reasons that they are shaped differently and all of them have something in common with the original graphs of sine and cosine.
Resources.
Mrs. Kirch on the wonderful website of https://www.desmos.com/calculator/hjts26gwst
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