Graphing Tangent, Cotangent, Secant, and Cosecant
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Key Questions

I am assuming that you are using a graphing calculator.
To graph
#sec x# , type in#1/(cos x)# and graph.graph{1/(cos x) [6.28, 6.28, 5, 5]}
Please note that the xaxis is in radians (I switched it to
#[2 pi, 2 pi])# . I would suggest using Zoom Trig (ZTrig) in the Zoom menu.To graph
#csc x# , type in#1/(sin x)# graph{1/(sin x) [6.28, 6.28, 5, 5]}
Finally, to graph
#cot x# , type in#i/(tan x)# graph{1/(tan x) [6.28, 6.28, 5, 5]}

secant graph is a cosine graph that is inversed

The asymptotes of y=cot(x) occur at whole number multiples of Ï€, or whole number multiples of 180Â°. Why?
For me, the best way to think about cotangent is to think about tangent, and the best way to think about tangent is by thinking about slope . In other words, the tangent of an angle is the slope that (terminal side) angle makes with the xaxis. As angle gets closer to 90Â°, slope gets larger and larger and eventually exceeds any finite number. So the asymptotes of tangent occur at the vertical positions: 90Â°, 270Â°, . . .
Cotangent, being the reciprocal of tangent, is the reciprocal slopein some sense, the rate of change of x versus y. Thus, the asymptotes of the cotangent will occur at the horizontal positions: 0Â°, 180Â°, 360Â° . . . , which are the whole number multiples of Ï€.