Real Arccosine Function

Theorem: Injectivity of the Cosine Function

The restriction of the real cosine function on the interval $[0;\pi ]$ is injective on its image and thus admits an inverse function.

PROOF

TODO

Definition: Real Arccosine Function

The real arccosine function is the inverse function of the restriction of the real cosine function on the interval $[0;\pi ]$.

NOTATION

$$\arccos {\cos }^{-1}$$

Note: Domain of the Real Arccosine Function

The domain of the real arccosine function is the interval $[-1;+1]$.

Note: Image of the Real Arcsine Function

The image the real arccosine function is the interval $[0;\pi ]$.