Definition:Inverse Cotangent/Real/Arccotangent

Definition
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

 * Definition:Cotangent

Other inverse trigonometrical ratios

 * Definition:Arcsine
 * Definition:Arccosine
 * Definition:Arctangent
 * Definition:Arcsecant
 * Definition:Arccosecant