# Cotangent Function
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/cotangent-function/
Fetched from algebrica.org post 6600; source modified 2026-03-06T22:27:34.
## Cotangent function
The cotangent function \\( f(x) = \\cot(x) \\) assigns to each angle \\( x \\), expressed in radians, its corresponding [cotangent](../tangent-and-cotangent.md) value. Its graph is a periodic curve with a period of \\( \\pi \\) and features vertical [asymptotes](../asymptotes.md) where the sine of \\( x \\) equals zero, specifically at \\( x = k\\pi \\) for \\( k \\in \\mathbb{Z} \\). The function \\( f(x) = \\cot(x) \\) has a [domain](../determining-the-domain-of-a-function.md) of all real numbers except these points, and its range is all real numbers.
- Domain: \\( { x \\in \\mathbb{R} : 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, \\( \\cot(-x) = -\\cot(x) \\)
* * *
- The cotangent of \\( x \\) is defined as the ratio between the [cosine and sine](../sine-and-cosine.md) of the angle \\( x \\).
\\[
\\cot(x) = \\frac{\\cos(x)}{\\sin(x)}
\\]
* * *
- Roots: \\( x = \\frac{\\pi}{2} + \\pi n, \\quad n \\in \\mathbb{Z} \\)
- Fundamental root: \\( x = \\frac{\\pi}{2} \\)
* * *
- Notable limits:
\\[
\\lim\\limits\_{x \\to 0} x \\cot(x) = 1
\\]
\\[
\\lim\_{x\\to0^+} \\cot(x) = +\\infty \\quad \\text{and} \\quad \\lim\_{x\\to0^-} \\cot(x) = -\\infty
\\]
* * *
- The function is [continuous](../continuous-functions.md) and differentiable on its domain.
- [Derivative](../derivatives.md):
\\[
\\frac{d}{dx} \\cot(x) = -\\csc^2(x)
\\]
* * *
- [Indefinite integral](../indefinite-integrals.md):
\\[
\\int \\cot(x) dx = \\ln |\\sin(x)| + c
\\]
##### A comprehensive overview of trigonometric integrals, together with the most useful transformation and substitution techniques for handling more complex cases, is available in the page on [trigonometric function integrals](../integral-of-trigonometric-functions.md).
* * *
- An alternative form of the function \\( \\cot(x) \\) using imaginary numbers is given by Euler’s formula. Here, \\( e^{ix} \\) is the [exponential function](../exponential-function.md) with base \\( e \\) and \\( i \\) is the [imaginary](../complex-numbers.md) unit. By expressing sine and cosine as
\\[
\\sin(x) = \\frac{e^{ix} - e^{-ix}}{2i} \\quad \\text{and} \\quad \\cos(x) = \\frac{e^{ix} + e^{-ix}}{2}
\\]
we obtain the cotangent function as
\\[
\\cot(x) = \\frac{\\cos(x)}{\\sin(x)} = i \\frac{e^{ix} + e^{-ix}}{e^{ix} - e^{-ix}}.
\\]
