Definition:Inverse Tangent/Real/Arctangent

Definition
From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From the same source, we also have that:
 * $\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
 * $\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$

Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

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

This function is called arctangent of $x$ and is written $\arctan x$.

Thus:
 * The domain of $\arctan x$ is $\R$
 * The image of $\arctan x$ is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Also see

 * Definition:Tangent Function

Other inverse trigonometrical ratios

 * Definition:Arcsine
 * Definition:Arccosine
 * Definition:Arccotangent
 * Definition:Arcsecant
 * Definition:Arccosecant