In tangent's ratio in the unit circle it's y/x. When you're trying to find the asymptotes for it you have to look where x would be undefined which means that the x has to equal 0. You would have to find the ordered pairs in which 0 is in the x spot, that would be in the 90 & 270 degrees. When you're going to put the asymptotes it would be on pi/2 & 3pi/2. Therefore the asymptotes would be at the 2 & 3 "quadrant", that will make the graph start from the bottom and make its way to the top without touching the asymptotes, being an uphill graph.
For cotangent the signs on the graph would be the same but the only thing that will make the difference would be the ratio, you now have the ratio of x/y. That means that you have to find the degrees in he unit circle where y will be 0. That could be found in 180 & 360 degrees. The asymptotes will shift from what they were in tangent. The asymptotes would be from the 3rd and 4th quadrants which will start of with the positive and then go down to negative. The graph will now be a downhill graph.
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