Definition:Inverse Cosine
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:
- $\inv \cos x := \set {y \in \R: \map \cos y = x}$
where $\map \cos y$ 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
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 $\closedint {-1} 1$.
This function is called the arccosine of $x$.
Thus:
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.