# Definition:Inverse Cotangent/Real

## Definition

Let $x \in \R$ be a real number such that $-1 \le x \le 1$.

The inverse cotangent of $x$ is the multifunction defined as:

$\cot^{-1} \left({x}\right) := \left\{{y \in \R: \cot \left({y}\right) = x}\right\}$

where $\cot\left({y}\right)$ is the cotangent of $y$.

### Arccotangent

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 $\openint 0 \pi$.