Definition:Inverse Tangent/Real

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Definition

Let $x \in \R$ be a real number.

The inverse tangent of $x$ is the multifunction defined as:

$\tan^{-1} \left({x}\right) := \left\{{y \in \R: \tan \left({y}\right) = x}\right\}$

where $\tan \left({y}\right)$ is the tangent of $y$.


Arctangent

Arctangent Function

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

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 $g: \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right) \to \R$ be the restriction of $\tan x$ to $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ 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 $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.