Tangent and cotangent are both positive in quadrant 1, negative in quadrant 2, positive in quadrant 3, and negative in quadrant 4. To start off with, we know that asymptotes exist whenever we divide by zero. For the graph of tangent, we need to look back at our ratios from the unit circle. Because tangent is equal to "y/x", tangent is going to have asymptotes wherever x equals zero (if "x" equals "0", we will receive an undefined answer). If we look at the graph, we know that the x value equals zero at pi/2 and 3pi/2. These two points will be the points of the asymptotes. With these asymptotes, we know that the graph has to be positive in quadrant 1 and quadrant 3 (ASTC) while it is negative in quadrant two and four. This outcome can only take place if the graph is positive in quadrant one, negative in quadrant two, so on and so forth. This makes the tangent graph uphill!
Cotangent is going to have asymptotes wherever sine equals zero. Yet again we need to look back at our unit circle ratios. Cotangent is positive in quadrant 1, negative in quadrant 2, positive in quadrant 3, and negative in quadrant 4. We also know that cotangent is equals to "y/x" (could be replaced with cos/sin). By looking at the graph, we see that this leads us to zero, pi, and 2pi. These values will become our asymptotes (that is where cotangent will become undefined). Because of the asymptotes and the sin graph, we will have to create a graph that goes from positive to negative to positive to negative. This can only be achieved one way with these asymptotes. this makes the normal cotangent graph downhill!
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