Definition:Inverse Tangent/Real/Arctangent

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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 $\arctan x$.


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


There exists the popular but misleading notation $\tan^{-1} x$, which is supposed to denote the inverse tangent function.

However, note that as $\tan x$ is not an injection, it does not have a well-defined inverse.

The $\arctan$ 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 $\tan^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname {Tan}^{-1}$ (with a capital $\text T$) is taken to mean the same as $\arctan$.

In computer software packages, the notation $\operatorname {atan}$ or $\operatorname {atn}$ can sometimes be found.

Some sources hyphenate: arc-tangent.

Also see

  • Results about inverse tangent can be found here.

Other inverse trigonometrical ratios