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1). Why is a "normal" tangent graph uphill but a "normal" cotangent graph is downhill?

The reason for this is that

tangent and cotangent have different ratios that cause for the asymptotes to be in different places.

- The ratio for tangent in terms of sine and cosine is sine/cosine. As a result, tangent will have asymptotes whenever and wherever cosine is on the x- axis (when cosine is equal to zero).
- In order for tangent to follow its ASTC pattern (positive, negative, positive, negative), the uphill shape shown above occurs in accordance with its asymptotes.

- Cotangent, on the other hand has asymptotes wherever sine is equal to zero on the y-axis because the trig ratio is cosine/sine.
- The difference in the asymptote location causes for the differences in shape. Since cotangent and tangent follow the same ASTC pattern, they both are positive is quadrants I and III and negative in II and IV. In order for cotangent to follow this pattern, it must be downhill.

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