# Definition:Inverse Cotangent/Real/Arccotangent

## Definition

Arccotangent Function

From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$.

From the same source, we also have that:

$\cot x \to + \infty$ as $x \to 0^+$
$\cot x \to - \infty$ as $x \to \pi^-$

Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.

This function is called arccotangent of $x$ and is written $\arccot x$.

Thus:

The domain of $\arccot x$ is $\R$
The image of $\arccot x$ is $\openint 0 \pi$.

## Caution

There exists the a popular but misleading notation $\cot^{-1} x$, which is supposed to denote the inverse cotangent function.

However, note that as $\cot x$ is not an injection, it does not have an inverse.

The $\arccot$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\cot^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname {Cot}^{-1}$ (with a capital $\text C$) is taken to mean the same as $\arccot$, although this can also be confusing due to the visual similarity between that and the lowercase $\text c$.

## Also see

• Results about inverse cotangent can be found here.