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

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

Definition

Domain and range

Where $\sin x = 0$, at $x = n\pi$ (with $n$ an integer), the function is undefined. Its domain is all real numbers except these points, and its range is all real numbers: like the tangent, its values are unbounded above and below.

Symmetry and period

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

Asymptotes and limits

There is a vertical asymptote at each $x = n\pi$. As $x \to 0^{+}$ (from the right), $\cot x \to +\infty$; as $x \to \pi^{-}$ (from the left), $\cot x \to -\infty$.

Behavior and zeros

On each interval $(n\pi,\ (n+1)\pi)$ the function is strictly decreasing, falling from $+\infty$ to $-\infty$. Its derivative $\dfrac{d}{dx}\cot x = -\dfrac{1}{\sin^2 x} = -\csc^2 x$ is always negative, confirming the decrease. Its zeros occur where $\cos x = 0$, at $x = \dfrac{\pi}{2} + n\pi$.

Here are some specific values.

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

Shape of the graph

On 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. This shape repeats under a shift of $\pi$, and each zero $x = \dfrac{\pi}{2} + n\pi$ sits exactly at the center of its interval.

Relation to other functions

It relates to the tangent by $\cot x = \tan\left(\dfrac{\pi}{2} - x\right)$.

Applications

The cotangent appears in surveying and triangulation, and in complex analysis and series, such as the partial-fraction expansion of $\pi \cot \pi x$.