Definition:Arccosine
Definition
From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\left[{1 \,.\,.\, \pi}\right]$.
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 $\left[{0 \,.\,.\, \pi}\right]$.
Caution
There exists the a 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 an 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 $\arccos$, although this can also be confusing due to the visual similarity between that and the lower case $\text{c}$.
Sources
- K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.5 \ (3)$
