Why is a "normal" tangent graph uphill, but a "normal" tangent graph downhill? Use unit circle ratios to explain.
The main reason why their graphs are different is because of their asymptotes and their location. The graphs are based on their Unit Circle ratios, and the graphs can't touch the asymptotes. The asymptotes themselves are based on the Unit Circle.
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| Both tangent and cotangent have their asymptotes based on their Unit Circle ratios. The ratio for tangent is y/x and in order to how and asymptote x has to 0 so it can be undefined. If we look at the graph, we see that 90 degrees and 270 degrees has the ratio to be undefined. 90 and 270 degrees in radians are pi/2 and 3pi/2, and those are the asymptotes. The same thing for cotangent except that the ratios are switched, it's x/y. So in order for cotangent to have asymptotes, y must be 0 and 180 and 360 degrees have those points. The asymptotes for cotangent, in radians, is pi and 2pi. |
Tangent:
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| If we look at it already graphed we can see that 2pi and 3pi/2 are the asymptotes and the graphs can't touch them. Also, based on the Unit Circle, the four quadrants and if they are positive and negative. Both quadrants 1 and 3 for tangent is positive so it's above the x-axis. Quadrants 2 and 4 are negative so the graph is below the x-axis. |
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Cotangent:
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| The asymptotes for cotangent is pi and 2pi and just like tangent, the graphs can't touch them. The Unit Circle and it's four quadrants are just like tangent, 1 and 3 is positive and 2 and 4 are negative. Expect that the graphs of cotangent will not be the same as tangent because the asymptotes make the graph go downhill to follow the positive- negative of the quadrants. |
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