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 $\openint 0 \pi$.

Also see

 * Definition:Cotangent

Other inverse trigonometrical ratios

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