y = \cot x

Graph of the Cotangent Function $y = \cot x$

The cotangent function $y = \cot x$ is the trigonometric function defined as the cosine divided by the sine¹. It is also the reciprocal of the tangent.

Definition

\cot x = \frac{\cos x}{\sin x} = \frac{1}{\tan x}

Domain and range

The domain is every real number with $x \neq n\pi$
The range is all real numbers
The period is $\pi$
It is an odd function

As with the tangent, the values have no upper or lower bound.

Symmetry and period

Since $\cot(-x) = -\cot x$ it is an odd function with point symmetry about the origin. The period is $\pi$ , matching the tangent and shorter than the $2\pi$ of the sine and cosine.

Asymptotes and limits

There is a vertical asymptote at each $x = n\pi$ .

Approach	$\cot x$
$x \to 0^{+}$	$+\infty$
$x \to \pi^{-}$	$-\infty$

Monotonicity and zeros

The derivative is as follows.

\frac{d}{dx}\cot x = -\frac{1}{\sin^2 x} = -\csc^2 x

It is always negative, so the function decreases monotonically on each interval $n\pi < x < (n+1)\pi$ . The zeros are the solutions of $\cos x = 0$ , namely $x = \dfrac{\pi}{2} + n\pi$ .

Notable values

$x$	$\cot x$
$\dfrac{\pi}{6}$	$\sqrt{3}$
$\dfrac{\pi}{4}$	$1$
$\dfrac{\pi}{3}$	$\dfrac{1}{\sqrt{3}}$
$\dfrac{\pi}{2}$	$0$

Shape of the graph

Take a single interval such as $0 < x < \pi$ . The curve starts near $+\infty$ at the left asymptote, crosses $0$ at $x = \dfrac{\pi}{2}$ , and descends toward $-\infty$ at the right asymptote. That shape repeats under a shift of $\pi$ , and each zero sits exactly at the center of its interval.

Comparison with the tangent

Item	$\tan x$	$\cot x$
Asymptotes	$\dfrac{\pi}{2} + n\pi$	$n\pi$
Zeros	$n\pi$	$\dfrac{\pi}{2} + n\pi$
Behavior on one branch	increasing	decreasing
Derivative	$\sec^2 x$	$-\csc^2 x$

The two are tied by $\cot x = \tan\left( \dfrac{\pi}{2} - x \right)$ , so the cotangent can also be read as the tangent reflected left to right and shifted by $\dfrac{\pi}{2}$ .

Applications

The relation between angles and lengths in surveying and triangulation
The partial-fraction expansion $\pi\cot\pi x$ in complex analysis
The antiderivative of $\csc^2 x$ in integral calculus

Trigonometric functions, Wikipedia

Graph of the Cotangent Function y=cot⁡xy = \cot xy=cotx