The 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 $[0;\pi ]$.

Properties

Theorem: Continuity of the Real Arccosine Function

The Real Arccosine Function is continuous.

PROOF

TODO

Theorem: Derivative of the Real Arccosine Function

The Real Arccosine Function is differentiable on the interval $(-1;+1)$ with

$$(\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$$

PROOF

TODO

Theorem: Antiderivatives of the Real Arccosine Function

The Antiderivatives of the Real Arccosine Function are given by

$$\int \arccos (x)\mathrm{dx}=x\arccos (x)-\sqrt{1-x^2}+\text{const}$$

PROOF

TODO