Graph of the Cotangent Function

−π −2π −3π π 2π 3π 10 9 8 7 6 5 4 3 2 1 −1 −2 −3 −4 −5 −6 −7 −8 −9 −10

The cotangent function is undefined at all integral multiples of π: $\dots - 3 π - 2 π - π 0 π 2 π 2 π \dots$

Because

the period of the cotangent function is π, and
consecutive integral multiples of π differ by π,

the entire graph of

y = \cot x

can be obtained by

graphing $y = \cot x$ on the open interval $(0, π)$ and then
copying that graph shifted to the right and to the left by integral multiples of π.

First, graph $y = \cot x$ on the open interval $(0, π)$ .

Use values of x at which exact cotangent function values are known to make a table.

`x`	$\cot x$
$\frac{π}{6}$	$\sqrt{3} ≅ 1.732$
$\frac{π}{4}$	1
$\frac{π}{3}$	$\frac{\sqrt{3}}{3} ≅ 0.577$
$\frac{π}{2}$	0
$\frac{2 π}{3}$	$- \frac{\sqrt{3}}{3} ≅ - 0.577$
$\frac{3 π}{4}$	−1
$\frac{5 π}{6}$	$- \sqrt{3} ≅ - 1.732$

Plot the points.

Find the value of the cotangent function at $0.05 π$ to see what happens to the graph when x is slightly greater than 0: $\cot 0.05 π ≅ 6.314$ Plot the point $(0.05 π, 6.314)$ .

Find the value of the cotangent function at $0.95 π$ to see what happens to the graph when x is slightly less than π: $\cot 0.95 π ≅ - 6.314$ Plot the point $(0.95 π, - 6.314)$ .

The line $x = 0$ and the line $x = π$ are vertical asymptotes for the graph.

Sketch from left to right a smooth curve through the points.

Copy the graph, shifting it to the left and to the right by integral multiples of π. Copy the asymptotes, too. The result is the graph of the cotangent function.