The Real Arcsine Function

Theorem: Injectivity of the Real Sine Function

The restriction of the Real Sine Function on the interval $\left[-\frac{\pi }{2};\frac{\pi }{2}\right]$ is injective on its image and thus admits an inverse function.

PROOF

TODO

Definition: Real Arcsine Function

The real arcsine function is the inverse function of the restriction of the Real Sine Function on the interval $\left[-\frac{\pi }{2};\frac{\pi }{2}\right]$.

NOTATION

$$\arcsin {\sin }^{-1}$$

Note: Domain of the Real Arcsine Function

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

Note: Image of the Real Arcsine Function

The image the real arcsine function is the interval $\left[-\frac{\pi }{2};\frac{\pi }{2}\right]$.

Properties

Theorem: Continuity of the Real Arccosine Function

The Real Arcsine Function is continuous.

PROOF

TODO

Theorem: Derivative of the Real Arcsine Function

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

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

PROOF

TODO

Theorem: Antiderivatives of the Real Arcsine Function

The Antiderivatives of the Real Arcsine Function are given by

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

PROOF

TODO