Real Trigonometric Functions

The Real Sine Function

Theorem: Convergence of the Sine Power Series

The real power series $\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}$ is convergent for all $x\in \mathbb{R}$.

PROOF

TODO

Definition: Real Sine Function

The real sine function is the real analytic function $\sin :\mathbb{R}\to \mathbb{R}$ defined by the real power series $\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}$.

$$\sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$$

NOTATION

$$\sin x\sin (x)$$

Theorem: Parity of the Real Sine Function

The real sine function is odd:

$$\sin (-x)=-\sin (x)$$

PROOF

TODO

Theorem: Image of the Real Sine Function

The image of the real sine function is $[-1;1]$.

$$\sin (\mathbb{R})=[-1;1]$$

PROOF

TODO

Theorem: Periodicity of the Real Sine Function

The real sine function has a period of $2\pi$. More generally,

$$\sin (x+2k\pi )=\sin (x)\forall k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Antiperiodicity of the Real Sine Function

The real sine function has an antiperiod of $\pi$. More generally,

$$\sin (x+(2k+1)\pi )=-\sin (x)k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Continuity of the Real Sine Function

The real sine function is continuous on $\mathbb{R}$.

PROOF

TODO

Theorem: Derivative of the Real Sine Function

The derivative of the real sine function is the real cosine function.

$$(\sin x{)}^{\prime }=\cos x$$

PROOF

$$\begin{array}{rl}(\sin x{)}^{\prime }& =\underset{\Delta x\to 0}{\lim }\frac{\sin (x+\Delta x)-\sin x}{\Delta x}=\underset{\Delta x\to 0}{\lim }\frac{2\sin \frac{\Delta x}{2}\cos \frac{2x+\Delta x}{2}}{\Delta x}\\ & =\underset{\Delta x\to 0}{\lim }\frac{\sin \frac{\Delta x}{2}}{\frac{\Delta x}{2}}\underset{\Delta x\to 0}{\lim }\cos \left(x+\frac{\Delta x}{2}\right)=1\cdot \cos x=\cos x\end{array}$$

The Real Cosine Function

Theorem: Convergence of the Cosine Power Series

The real power series $\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$ is convergent for all $x\in \mathbb{R}$.

PROOF

TODO

Definition: Real Cosine Function

The real cosine function is the real analytic function $\cos :\mathbb{R}\to \mathbb{R}$ defined by the real power series $\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$.

$$\cos (x)=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$$

NOTATION

$$\cos x\cos (x)$$

Theorem: Parity of the Real Cosine Function

The real cosine function is even:

$$\cos (-x)=\cos (x)$$

PROOF

TODO

Theorem: Image of the Real Cosine Function

The image of the real cosine function is $[-1;1]$.

$$\cos (\mathbb{R})=[-1;1]$$

PROOF

TODO

Theorem: Periodicity of the Real Cosine Function

The real cosine function has a period of $2\pi$. More generally,

\cos (x + 2k\pi) = \cos(x) \qquad \forall k\in\mathbb{Z}$$ >[!PROOF]- > >TODO >

Theorem: Antiperiodicity of the Real Cosine Function

The real cosine function has an antiperiod of $\pi$. More generally,

$$\cos (x+(2k+1)\pi )=-\cos (x)k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Continuity of the Real Cosine Function

The real cosine function is continous on $\mathbb{R}$.

PROOF

TODO

Theorem: Derivative of the Real Cosine Function

The real cosine function is differentiable and its derivative is the negative real sine function.

$$(\cos x{)}^{\prime }=-\sin x$$

PROOF

$$(\cos x{)}^{\prime }=\sin {\left(\frac{\pi }{2}-x\right)}^{\prime }={\left(\frac{\pi }{2}-x\right)}^{\prime }\cos \left(\frac{\pi }{2}-x\right)=-\cos \left(\frac{\pi }{2}-x\right)=-\sin x$$

The Real Tangent Function

Definition: Real Tangent Function

The real tangent function is defined as the ratio of the real sine function to the real cosine function.

$$\tan (x)\stackrel{\text{def}}{=}\frac{\sin (x)}{\cos (x)}$$

NOTE

The domain of the real tangent function is $\{x\in \mathbb{R}\mid x\ne \frac{\pi }{2}+k\pi ,k\in \mathbb{Z}\}$ because $\cos (\frac{\pi }{2}+k\pi )=0$ for all $k\in \mathbb{Z}$.

NOTATION

$$\tan (x)\mathrm{tg}(x)$$

Theorem: Image of the Real Tangent Function

The image of the real tangent function is $(-\infty ;+\infty )$.

PROOF

TODO

Theorem: Periodicity of the Real Tangent Function

The real tangent function has a period of $\pi$. More generally,

$$\tan (x+k\pi )=\tan (x)\forall k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Continuity of the Real Tangent Function

The real tangent function is continuous.

PROOF

TODO

Theorem: Derivative of the Real Tangent Function

The real tangent function is differentiable and its derivative is the recirpocal of the square of the real cosine function:

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

PROOF

$$(\tan x{)}^{\prime }={\left(\frac{\sin x}{\cos x}\right)}^{\prime }=\frac{(\sin x{)}^{\prime }\cos x-(\cos x{)}^{\prime }\sin x}{{\cos }^{2}x}=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}=\frac{1}{{\cos }^{2}x}$$

The Real Cotangent Function

Definition: Real Cotangent Function

The real cotangent function is defined as the ratio of the real cosine function to the real sine function.

$$\cot (x)\stackrel{\text{def}}{=}\frac{\cos (x)}{\sin (x)}$$

NOTE

The domain of the real cotangent function is $\{x\in \mathbb{R}\mid x\ne k\pi ,k\in \mathbb{Z}\}$ because $\sin (k\pi )=0$ for all $k\in \mathbb{Z}$.

NOTATION

$$\cot (x)\mathrm{ctg}(x)\mathrm{cotg}(x)$$

Theorem: Image of the Real Cotangent Function

The image of the real cotangent function is $(-\infty ;+\infty )$.

PROOF

TODO

Theorem: Periodicity of the Real Cotangent Function

The real cotangent function has a period of $\pi$. More generally,

$$\cot (x+k\pi )=\cot (x)\forall k\in \mathbb{Z}$$

PROOF

TODO

Theorem: Continuity of the Real Cotangent Function

The real cotangent function is continuous.

PROOF

TODO

Theorem: Derivative of the Real Cotangent Function

The real cotangent function is differentiable and its derivative is the negative reciprocal of the square of the real sine function:

$$(\cot x{)}^{\prime }=-\frac{1}{{\sin }^{2}x}=-1-{\cot }^{2}x$$

PROOF

$$(\cot x{)}^{\prime }={\left(\frac{\cos x}{\sin x}\right)}^{\prime }=\frac{(\cos x{)}^{\prime }\sin x-(\sin x{)}^{\prime }\cos x}{{\sin }^{2}x}=\frac{-{\sin }^{2}x-{\cos }^{2}x}{{\sin }^{2}x}=\frac{-({\sin }^{2}x+{\cos }^{2}x)}{{\sin }^{2}x}=\frac{-1}{{\sin }^{2}x}$$

Theorem: Antiderivatives of $\cot$

The antidirivatives of the real cotangent are given by

$$\int \cot x\mathrm{dx}=\ln |\sin x|+C$$

PROOF

TODO