Category:Arctangent Function

From ProofWiki
Jump to navigation Jump to search

This category contains results about Arctangent Function.

Arctangent Function

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