Definition:Inverse Cosine/Real

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Definition

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:

$\map {\cos^{-1} } x := \set {y \in \R: \map \cos y = x}$

where $\map \cos y$ is the cosine of $y$.


Arccosine

Real Arccosine Function

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:

The domain of arccosine is $\closedint {-1} 1$
The image of arccosine is $\closedint 0 \pi$.


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.