Cotangent Function

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: $x = \frac{\pi}{2} + \pi n , n \in \mathbb{Z}$
• Fundamental root: $x = \frac{\pi}{2}$

• Notable limits:

  \[\underset{x \rightarrow 0}{lim} x cot ⁡ ( x ) = 1\] \[\underset{x \rightarrow 0^{+}}{lim} cot ⁡ ( x ) = + \infty \text{and} \underset{x \rightarrow 0^{-}}{lim} cot ⁡ ( x ) = - \infty\]

• The function is continuous and differentiable on its domain.
• Derivative: \(\frac{d}{d x} cot ⁡ ( x ) = - csc^{2} ⁡ ( x )\)

• Indefinite integral: \(\int cot ⁡ ( x ) d x = ln ⁡ | sin ⁡ ( x ) | + c\)

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, $e^{i x}$ is the exponential function with base $e$ and $i$ is the imaginary unit. By expressing sine and cosine as \(sin ⁡ ( x ) = \frac{e^{i x} - e^{- i x}}{2 i} \text{and} cos ⁡ ( x ) = \frac{e^{i x} + e^{- i x}}{2}\) we obtain the cotangent function as \(cot ⁡ ( x ) = \frac{cos ⁡ ( x )}{sin ⁡ ( x )} = i \frac{e^{i x} + e^{- i x}}{e^{i x} - e^{- i x}} .\)