Definition:Inverse Cosine/Arccosine
Definition
Real Numbers
From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$.
From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.
Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$.
Thus:
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 $z^2 - 1$.
Terminology
There exists the popular but misleading notation $\cos^{-1} x$, which is supposed to denote the inverse cosine function.
However, note that as $\cos x$ is not an injection (even though by restriction of the codomain it can be considered surjective), it does not have a well-defined inverse.
The $\arccos$ 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 $\cos^{-1} x$, which (as pointed out) can be confusing and misleading.
Sometimes, $\operatorname{Cos}^{-1}$ (with a capital $\text C$) is taken to mean the same as $\arccos$.
However, this can also be confusing due to the visual similarity between that and the lower case $\text c$.
Some sources hyphenate: arc-cosine.
Symbol
The symbol used to denote the arccosine function is variously seen as follows:
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccosine function is $\arccos$.
A variant symbol used to denote the arccosine function is $\operatorname {acos}$.
Also known as
The word arccosine can also be seen as two words: arc cosine.