Why is a “normal” tangent graph uphill, but a “normal” tangent graph downhill? Use unit circle ratios to explain.
We know that the ratio for tangent is y/x. This means the asymptote is where x is equal to 0 (making the ratio undefined). The x-value is equal to O at (0,1) and (0,-1) which is pi/2 and 3pi/2, which could also be referred to as -pi/2. When we draw these asymptotes located at pi/2 and -pi/2 on
a graph, the period consists of Q4 and Q1. According to ASTC, Tan is negative in Q4 and positive
in Q1. Because of this, the only possible way to draw this is with an uphill graph. It will begin below the x-axis in Q4 and then move above the x-axis when it enters into Q1. For cot, which is x/y, the asymptote is where y is equal to 0. This
occurs at (1,0) and (-1,0), which is 0 and pi. When we draw the asymptotes, the period consists of Q1 and Q2. Because cot is pos. in Q1 and neg. in Q2, the shift is from
pos to neg. The only way to draw this downhill. The cot graph will begin near the asymptote above the x-axis and then go downhill to below the x-axis when it reaches Q2. Through this reasoning, a normal tangent graph is uphill while a normal cotangent graph is downhill.
The image below will help clarify the difference between tan and cot in reference to uphill or downhill.