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 value. Its graph is a periodic curve with a period of $\pi$ and features vertical asymptotes 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 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 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 and differentiable on its domain.
- Derivative:
\frac{d}{dx} \cot(x) = -\csc^2(x)
\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.
- An alternative form of the function $\cot(x)$ using imaginary numbers is given by Euler’s formula. Here, $e^{ix}$ is the exponential function with base $e$ and $i$ is the imaginary 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}}.