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The Real Arcsine Function#

Theorem: Injectivity of the Real Sine Function

The restriction of the real sine function on \(\displaystyle \left[-\frac{\pi}{2}; \frac{\pi}{2}\right]\) is injective on its image.

Proof

TODO

Definition: Real Arcsine Function

The real arcsine function is the inverse function of the restriction of the real sine function on \(\displaystyle \left[-\frac{\pi}{2}; \frac{\pi}{2}\right]\).

Notation

\[ \arcsin \qquad \sin^{-1} \]

Note: Domain of the Real Arcsine Function

The domain of the real arcsine function is \([-1; +1]\).

Note: Image of the Real Arcsine Function

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

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 \((-1;+1)\) with

\[ (\arcsin x)' = \frac{1}{\sqrt{1 - x^2}} \]
Proof

TODO

Theorem: Antiderivatives of the Real Arcsine Function

The antiderivatives of the real arcsine function are given by

\[ \int \arcsin(x) \mathop{\mathrm{d}x} = x \arcsin(x) + \sqrt{1 - x^2} + \text{const} \]
Proof

TODO

The Real Arccosine Function#

Theorem: Injectivity of the Cosine Function

The restriction of the real cosine function on \([0;\pi]\) is injective on its image.

Proof

TODO

Definition: Real Arccosine Function

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

Notation
\[ \arccos \qquad \cos^{-1} \]

Note: Domain of the Real Arccosine Function

The domain of the real arccosine function is \([-1; +1]\).

Note: Image of the Real Arcsine Function

The image the real arccosine function is \([0; \pi]\).

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 \((-1;+1)\) with

\[ (\arccos x)' = - \frac{1}{\sqrt{1 - x^2}} \]
Proof

TODO

Theorem: Antiderivatives of the Real Arccosine Function

The antiderivatives of the real arccosine function are given by

\[ \int \arccos (x) \mathop{\mathrm{d}x} = x \arccos (x) - \sqrt{1 - x^2} + \text{const} \]
Proof

TODO

The Real Arctangent Function#

Theorem: Injectivity of the Real Tangent Function

The restriction of the real tangent function on \(\displaystyle \left(-\frac{\pi}{2}; \frac{\pi}{2} \right)\) is injective.

Proof

TODO

Definition: Real Arctangent Function

The real arctangent function is the inverse function of the restriction of the real tangent function on \(\displaystyle \left(-\frac{\pi}{2}; \frac{\pi}{2} \right)\).

Notation

\[ \arctan \qquad \tan^{-1} \]

Note: Domain of the Real Arctangent Function

The domain of the real arctangent function is \(\mathbb{R}\).

Note: Domain of the Real Arctangent Function

The image of the real arctangent function is \(\displaystyle \left(-\frac{\pi}{2}; \frac{\pi}{2} \right)\).

Theorem: Continuity of the Real Arctangent Function

The real arctangent function is continuous.

Proof

TODO

Theorem: Derivative of the Real Arctangent Function

The real arctangent function is differentiable with

\[ (\arctan x)' = \frac{1}{x^2 + 1} \]
Proof

TODO

Theorem: Antiderivatives of the Real Arctangent Function

The antiderivatives of the real arctangent function are given by

\[ \int \arctan(x) \mathop{\mathrm{d}x} = x \arctan (x) - \frac{1}{2}\ln(1+ x^2) + \text{const} \]
Proof

TODO

The Real Arccotangent Function#

Theorem: Injectivity of the Real Cotangent Function

The restriction of the real tangent function is injective on \((0; \pi)\).

Proof

TODO

Definition: Real Arccotangent Function

The real arccotangent function is the inverse function of the restriction of the real tangent function on \((0; \pi)\).

Notation
\[ \operatorname{arccot} \qquad \cot^{-1} \]

Note: Domain of the Real Arctangent Function

The domain of the real arccotangent function is \(\mathbb{R}\).

Note: Domain of the Real Arctangent Function

The image of the real arccotangent function is \((0; \pi)\).

Theorem: Continuity of the Real Arccotangent Function

The real arccotangent function is continuous.

Proof

TODO

Theorem: Derivative of the Real Arccotangent Function

The real arccotangent function is differentiable with

\[ (\mathop{\mathrm{arccot}} x)' = -\frac{1}{x^2 + 1} \]
Proof

TODO

Theorem: Antiderivatives of the Real Arccotangent Function

The antiderivatives of the real arccotangent function are given by

\[ \int \operatorname{arccot}(x) \mathop{\mathrm{d}x} = x \operatorname{arccot}(x) + \frac{1}{2}\ln(1 + x^2) + \text{const} \]
Proof

TODO