Definition:Inverse Cosine

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Definition

Real Numbers

Let $x \in \R$ be a real number such that $-1 \le x \le 1$.

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

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

where $\cos \left({y}\right)$ is the cosine of $y$.


Complex Plane

Let $z \in \C$ be a complex number.

The inverse cosine of $z$ is the multifunction defined as:

$\cos^{-1} \left({z}\right) := \left\{{w \in \C: \cos \left({w}\right) = z}\right\}$

where $\cos \left({w}\right)$ is the cosine of $w$.


Arccosine

Real Numbers

Arccosine Function

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\left[{0 \,.\,.\, \pi}\right]$.

From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.


Therefore, let $g: \left[{0 \,.\,.\, \pi}\right] \to \left[{-1 \,.\,.\, 1}\right]$ be the restriction of $\cos x$ to $\left[{0 \,.\,.\, \pi}\right]$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\left[{-1 \,.\,.\, 1}\right]$.


This function is called arccosine of $x$ and is written $\arccos x$.


Thus:

  • The domain of $\arccos x$ is $\left[{-1 \,.\,.\, 1}\right]$
  • The image of $\arccos x$ is $\left[{0 \,.\,.\, \pi}\right]$.


Complex Plane

The principal branch of the complex inverse cosine function is defined as:

$\map \arccos z = \dfrac 1 i \, \map \Ln {z + \sqrt {z^2 - 1} }$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.


Also see