Introduction

The geometric construction of the cotangent from the unit circle is developed in tangent and cotangent. Here, the cotangent is treated as a real function of a real variable.

The cotangent function $f(x) = \cot(x)$ assigns to each angle $x,$ measured in radians, its corresponding cotangent value. Its graph is a periodic curve with period $\pi$ and has vertical asymptotes where the sine of $x$ vanishes, at $x = k\pi$ with $k \in \mathbb{Z}.$ The domain is the set of all real numbers except these points, and the range is all of $\mathbb{R}.$

The cotangent function drops sharply near its vertical asymptotes and crosses the x-axis at every half- integer multiple of π. y x π -π 0 2π

The cotangent is the ratio of cosine and sine so it diverges where the sine approaches zero and crosses the horizontal axis where the cosine vanishes:

$$\cot(x) = \frac{\cos(x)}{\sin(x)}$$

On the interval $(0, \pi)$ the curve descends from $+\infty$ near $x = 0$ to $-\infty$ near $x = \pi,$ crossing zero at $x = \pi/2.$

Properties

The following properties of the cotangent function follow from its definition as the ratio of cosine to sine.

  • Domain: $\{\ x \in \mathbb{R} \mid x \neq k\pi \ \text{ for all } k \in \mathbb{Z} \ \}$
  • Range: $y \in \mathbb{R}$
  • Periodicity: periodic in $x$ with period $\pi$
  • Parity: odd, with $\cot(-x) = -\cot(x)$
  • Monotonicity: decreasing on each interval $\left(k\pi, \pi + k\pi\right)$ with $k \in \mathbb{Z}$
  • Roots: $x = \frac{\pi}{2} + n\pi$ with $n \in \mathbb{Z}$
  • No root is an integer, since $\frac{\pi}{2} + n\pi$ is irrational for every $n \in \mathbb{Z}.$

Limits, derivatives, and integrals of the cotangent function

A remarkable limit describes the cotangent in a neighbourhood of the origin:

$$\lim_{x \to 0} x\cot(x) = 1$$

The behaviour near the asymptote at the origin is described by one-sided limits. As $x$ approaches $0$ from the right the sine is positive and tends to zero while the cosine tends to $1,$ so the function grows without bound,

$$\lim_{x \to 0^+} \cot(x) = +\infty$$

while from the left the sine is negative and the values diverge to negative infinity,

$$\lim_{x \to 0^-} \cot(x) = -\infty$$

The function is continuous and differentiable on its domain. Its derivative is:

$$\frac{d}{dx}\cot(x) = -\csc^2(x)$$

The indefinite integral is:

$$\int \cot(x) \ dx = \ln|\sin(x)| + c$$

A broader treatment of trigonometric integrals, with the transformation and substitution techniques for the more complex cases, is given in trigonometric function integrals.

The cotangent function can also be written using imaginary numbers. With $e^{ix}$ the exponential function of base $e$ and $i$ the imaginary unit, Euler's formula gives:

$$\cot(x) = \frac{i\left(e^{ix} + e^{-ix}\right)}{e^{ix} - e^{-ix}}$$

Inverse function

On its whole domain the cotangent is not injective, because the period $\pi$ makes it repeat every value in each branch. Restricted to the open interval $\left(0, \pi\right),$ where it is continuous and strictly decreasing, the cotangent is a bijection onto $\mathbb{R}$ and admits an inverse function, the arccotangent:

$$\mathrm{arccot} : \mathbb{R} \to \left(0, \pi\right)$$

On this restricted domain $\mathrm{arccot}(\cot(x)) = x,$ and $\cot(\mathrm{arccot}(y)) = y$ for every real $y.$