Definition:Inverse Cosine/Real/Arccosine

Definition
From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\left[{0 \,.\,.\, \pi}\right]$.

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 $\text{c}$.

Also see

 * Definition:Cosine

Other inverse trigonometrical ratios

 * Definition:Arcsine
 * Definition:Arctangent
 * Definition:Arccotangent
 * Definition:Arcsecant
 * Definition:Arccosecant