Inverse Real Trigonometric Functions

The Real Arcsine Function

Theorem: Injectivity of the Real Sine Function

The restriction of the real sine function on $\left[-\frac{\pi }{2};\frac{\pi }{2}\right]$ 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 $\left[-\frac{\pi }{2};\frac{\pi }{2}\right]$.

NOTATION

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

Note: Domain of the Real Arcsine Function

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

Note: Image of the Real Arcsine Function

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

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

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

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

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

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

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

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

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

The Real Arctangent Function

Theorem: Injectivity of the Real Tangent Function

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

NOTATION

$$\arctan {\tan }^{-1}$$

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 $\left(-\frac{\pi }{2};\frac{\pi }{2}\right)$.

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

$$(\arctan x{)}^{\prime }=\frac{1}{x^2+1}$$

PROOF

TODO

Theorem: Antiderivatives of the Real Arctangent Function

The antiderivatives of the real arctangent function are given by

$$\int \arctan (x)\mathrm{dx}=x\arctan (x)-\frac{1}{2}\ln (1+x^2)+\text{const}$$

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

$$\mathrm{arccot}{\cot }^{-1}$$

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

$$(\mathrm{arccot}x{)}^{\prime }=-\frac{1}{x^2+1}$$

PROOF

TODO

Theorem: Antiderivatives of the Real Arccotangent Function

The antiderivatives of the real arccotangent function are given by

$$\int \mathrm{arccot}(x)\mathrm{dx}=x\mathrm{arccot}(x)+\frac{1}{2}\ln (1+x^2)+\text{const}$$

PROOF

TODO