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

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

Properties

Theorem: Parity of the Real Cosine Function

The Real Cosine Function is ^even-function:

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

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 continuous on $\mathbb{R}$.

PROOF

TODO

Theorem: Derivative of the Real Cosine Function

The Real Cosine Function is differentiable everywhere and its derivative is the negative Real Sine Function.

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

PROOF

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