Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?
This picture represents the different quadrants and the graph of cosine. We graph cosine because we know based off of the unit circle ratios that tangent is undefined when cosine equals zero (tan=sin/cos). That means there are asymptotes where the graph of cosine hits the x-axis (pi/2, 3pi/2). Due to this, the graph of tangent must be broken between quadrants 1 and 2, it cannot continue because of that asymptote. However, it continues between quadrants 2 and 3, as there is no asymptote at pi (cosine does not touch the x-axis/equal zero). This means it can follow a downhill then uphill shape, giving the normal tangent graph an "uphill" look.
Cotangent is undefined when sin equals zero since the unit circle ratio of cotangent is (cos/sin). This means there are asymptotes where sin=0, a.k.a. where it hits the x-axis (0, pi, and 2pi). Since there is no asymptote between quadrants 1 and 2 (sine does not touch x-axis/does not equal zero), the graph of cotangent can be continuous. The only way we can graph this is by following a downhill pattern, giving cotangent a "downhill" look.
As seen by the images, the graph of tangent and cotangent differ because of the location of the asymptotes. In tangent, the pattern is "broken" by an asymptote, causing it to have an uphill look. Cotangent is not broken by asymptotes and thus ensues a downhill look.
REFERENCE: All graphs from Desmos.com and template from Mrs. Kirch. YOu can view more of her blogs here.
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