## Definitions

### General

The inverse cotangent function, in modern notation written as arccot(x), gives the angle θ, so that:

\[\cot \theta = x \]

Due to the periodical nature of the cotangent function, there are many angles θ that can give the same cotangent value (i.e. θ+π, θ+3π, etc.). As a result, it is impossible to define a single inverse function, unless the range of the return values is restricted, so that a one-to-one relationship between θ and cotθ can be established. Therefore, multiple branches of the arccot function can be defined. Commonly, the desired range of θ values spans between 0 and π. The branch of arccot, in that case, is called the principal branch.

### Series

The arccot function can be defined in a Taylor series form, like this:

\[ \begin{split} \textrm{arccot}\, x & = \frac{\pi}{2} - \arctan x =\\ & = \frac{\pi}{2} - \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1} = \\ & = \frac{\pi}{2}- x + \frac{x^3}{3} - \frac{x^5}{5} + \frac{x^7}{7} \cdots \end{split} \]

From the expanded form of the series, it can be seen that the higher terms become insignificant, for values of x close to zero, resulting in the following quite useful approximation:

\[ \textrm{arccot}\, x \approx \frac{\pi}{2} -x, \quad x \to 0, \quad \textrm{result x in radians} \]

### Properties

The derivative of the arccot function is:

\[ \left(\textrm{arccot}\, x\right)' = -\frac{1}{1+x^2}\]

The integral of the arccot function is given by:

\[ \int \textrm{arccot}\, x\, \mathrm{d}x = x\, \textrm{arccot}\, x +\frac{\ln\left(1+x^2\right)}{2} + C \]

The following properties are also valid for the arccot function:

\[ \begin{split} & \cot (\textrm{arccot}\, x) &= x \\ \\ & \sin (\textrm{arccot}\, x) &= \frac{1}{\sqrt{1+x^2}} \\ \\ & \cos (\textrm{arccot}\, x) &= \frac{x}{\sqrt{1+x^2}} \\ \\ & \tan (\textrm{arccot}\, x) &= \frac{1}{x} \\ \\ & \textrm{arccot} \left(-x\right) &= \pi- \textrm{arccot}\, x \\ \\ & \textrm{arccot}\, x &= \frac{\pi}{2} - \arctan x \\ \\ \end{split} \]