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

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

Properties

Theorem: Parity of the Real Sine Function

The Real Sine Function is ^odd-function:

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

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

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