\begin{definition}[Definition:Inverse Cosine/Real/Arccosine]
{{:Graph of Arccosine Function|Graph}}
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 '''arccosine of $x$''' and is written $\arccos x$.
Thus:
:The domain of $\arccos x$ is $\closedint {-1} 1$
:The image of $\arccos x$ is $\closedint 0 \pi$.
\end{definition}