Graph of cotangent: y = cot x

Updated 20 October 2026 · Values computed by the ihateDesmos engine

The cotangent graph is the reciprocal of tangent: cot x = cos x / sin x. It has period π, a vertical asymptote wherever sin x = 0, and every branch falls from +∞ to −∞.

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The graph of

y=cot(x)

for −7 ≤ x ≤ 7. Dots mark its roots.

Key features

Domain: All real x except x = nπ
Range: All real numbers
Period: π
Vertical asymptotes: x = nπ, including x = 0
Zeros: x = π/2 + nπ
Symmetry: Odd: cot(−x) = −cot x
y-intercept: None: x = 0 is an asymptote
Derivative: −1sin(x)²
Antiderivative: ln(|sin(x)|)+C

Derivative and antiderivative from the calculator’s algebra system. Domain, range and asymptotes are stated for real x.

What the graph of cot x looks like

Each branch lives between two consecutive multiples of π. It starts near +∞ just to the right of the left asymptote, falls through zero at the midpoint π/2 + nπ, and dives to −∞ at the right asymptote.

Cotangent is decreasing on every branch, the mirror image of tangent’s increasing branches. In fact cot x = tan(π/2 − x), so the cotangent graph is the tangent graph reflected in the y-axis and shifted by π/2.

The two functions trade places: where tan x is 0, cot x has an asymptote, and where tan x has an asymptote, cot x is 0.

Points of interest

Found numerically to about twelve digits in the window −7 ≤ x ≤ 7, then shown in exact form where one exists.

x-intercepts (roots): x = −4.7124, x = −1.5708, x = 1.5708, x = 4.7124
Local maxima: None in this window
Local minima: None in this window
Inflection points: (−3π/2, 0), (−π/2, 0), (π/2, 0), (3π/2, 0)

Table of values

A few points to plot, with exact values where they are simple.

Values of cot x
x	y = cot x
π/6	√3
π/4	1
π/3	√3/3
π/2	0
2π/3	−√3/3
3π/4	−1
5π/6	−√3

How to graph cot x by hand

1
Draw dashed asymptotes at x = 0 and x = π.
2
Plot (π/4, 1), (π/2, 0) and (3π/4, −1).
3
Draw a decreasing curve through them that rises along the left asymptote and falls along the right one.
4
Repeat the branch every π.

Transformations

How the graph moves when you change the formula. The original is cot x in blue.

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y = cot(2x) Period π/2: asymptotes every π/2.
y = -cot(x) Reflected in the x-axis, so each branch increases. This equals tan(x − π/2).
y = cot(x) + 1 Shifted up 1: the zeros move to where cot x = −1, at x = 3π/4 + nπ.