Definition:Inverse Cosine/Real/Arccosine

Definition
From Nature of Cosine Function, we have that $$\cos x$$ is continuous and strictly decreasing on the interval $$\left[{1 \,. \, . \, \pi}\right]$$.

From Basic Properties of Cosine Function, $$\cos \pi = -1$$ 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 C) is taken to mean the same as $$\arccos$$, although this can also be confusing due to the visual similarity of upper and lower case c.