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\documentclass{../../../lecture}
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\begin{document}
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\begin{satz}[Reihenentwicklung Sinus / Cosinus]
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Für alle $x \in \R$ gilt (absolut konvergente
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Potenzreihendarstellung)
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\[
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\cos(x) = \sum_{k=0}^{\infty} (-1)^{k}\frac{x^{2k}}{(2k)!} = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \ldots
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.\] und
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\[
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\sin(x) = \sum_{k=0}^{\infty} (-1)^{k}\frac{x^{2k+1}}{(2k+1)!} = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \ldots
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.\]
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\end{satz}
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\begin{proof}
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Die absolute Konvergenz folgt als Teilreihe der Exponentialreihe (als Majorante)
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Es gilt für $m \in \N_0$
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\[
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i^{n} = \begin{cases}
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1 & n = 4m \\
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i & n = 4m+1 \\
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-1 & n = 4m+2 \\
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-i & n = 4m+3
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\end{cases}
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.\] Es folgt
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\begin{align*}
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e^{ix} &= \sum_{n=0}^{\infty} \frac{(ix)^{n}}{n!} = \sum_{n=0}^{\infty} i^{n} \frac{x^{n}}{n!} \\
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&= \underbrace{\sum_{k=0}^{\infty} (-1)^{k} \frac{x^{2k}}{(2k)!}}_{\cos(x)} + i \underbrace{\sum_{k=0}^{\infty} (-1)^{k} \frac{x^{2k+1}}{(2k+1)!}}_{\sin(x)}
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.\end{align*}
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\end{proof}
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\begin{satz}[Restgliedabschätzung Sinus / Cosinus]
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Für $n \in \N_0$ gilt
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\[
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\cos(x) = \sum_{k=0}^{n} (-1)^{k} \frac{x^{2k}}{(2k)!} + R_{2n+2}(x)
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.\] und
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\[
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\sin(x)= \sum_{k=0}^{n} (-1)^{k} \frac{x^{2k+1}}{(2k+1)!} + R_{2n+3}(x)
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.\]
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mit
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\[
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|R_{2n+2}(x)| \le \frac{|x|^{2n+2}}{(2n+2)!} \text{ für } |x| \le 2n+3
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.\] bzw.
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\[
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|R_{2n+3}(x)| \le \frac{|x|^{2n+3}}{(2n+3)!} \text{ für } |x| \le 2n+4
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.\]
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\end{satz}
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\begin{proof}
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Es gilt
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\begin{align*}
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R_{2n+2}(x) &= \sum_{k=n+1}^{\infty} (-1)^{k} \frac{x^{2k}}{(2k)!} \\
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&= (-1)^{n+1} \frac{x^{2n+2}}{(2n+2)!}
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\left( \sum_{k=n+1}^{\infty} (-1)^{k-(n+1)}
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\frac{x^{2(k - (n+1))}}{(2k)! \frac{1}{(2n+2)!}}\right) \\
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&= (-1)^{n+1} \frac{x^{2n+2}}{(2n+2)!}
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\left( \sum_{k=0}^{\infty} (-1)^{k}
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\frac{x^{2k}(2n+2)!}{(2k+2n+2)!} \right)
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.\end{align*}
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Für $k \in \N$ setze
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\begin{align*}
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a_k :&= \frac{x^{2k}(2n+2)!}{(2k+2n+2)!}
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= \frac{x^{2k}}{(2n+3)(2n+4) \ldots (2k + 2n + 2)} \\
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a_{k-1} &= \frac{x^{2k-2}(2n+2)!}{(2k+2n)!}
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\intertext{damit}
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a_k &= a_{k-1} \cdot \frac{x^{2}}{(2k+2n+1)(2k+2n+2)}
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.\end{align*}
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Es gilt für $|x| \le 2n+3, k\ge 1$
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\[
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\frac{x^{2}}{(2k+2n+1)(2k+2n+2)} \le \frac{(2n+3)^{2}}{(2n+3)(2n+4)} < 1
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.\] $\implies$
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\[
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a_k \le \frac{(2n+3)^{k}}{(2n+4)^{k}} a_0 \quad a_0 = \frac{1}{(2n+2)!}
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.\] $\stackrel{\text{Leibniz}}{\implies}$
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\[
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\sum_{k=0}^{\infty} (-1)^{k} a_k
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.\] konvergent mit
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\[
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0 < \underbrace{\underbrace{1 - a_1}_{> 0} + \underbrace{a_2 - a_3}_{> 0}
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+ \underbrace{a_4 - \ldots}_{> 0}}_{< 1} < 1
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.\] $\implies$
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\[
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|R_{2n+2}(x)| \le \frac{|x|^{2n+2}}{(2n+2)!} \text{ für } |x| \le 2n+3
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.\] Genauso für $R_{2n+3}(x)$ (Sinus).
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\end{proof}
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\begin{lemma}
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Sinus und Cosinus Funktionen haben das folgende Verhalten
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\[
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\lim_{x \to 0} \frac{\sin(x)}{x} = 1
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.\]
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\[
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\lim_{x \to 0} \frac{\cos(x)-1}{x} = 0
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.\]
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\end{lemma}
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\begin{proof}
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\begin{align*}
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\left| \frac{\sin(x)}{x} - 1 \right|
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&= \left| \underbrace{1 - \frac{x^{2}}{3!} + \frac{x^{4}}{5!}}_{\frac{\sin(x)}{x}} - \ldots - 1\right| \\
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&= \left| x \sum_{k=1}^{\infty} (-1)^{k}\frac{x^{2k-1}}{(2k+1)!} \right| \\
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&\stackrel{|x| < 1}{\le |x|} \cdot \left| \sum_{k=1}^{\infty} \frac{1}{(2k+1)!} \right|
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\le |x| \cdot e
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.\end{align*} $\implies$
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\[
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\underbrace{\left| \frac{\sin(x)}{x} -1 \right|}_{\to 0}
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\le \underbrace{|x| \cdot e}_{\to 0}
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.\]
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genauso für $\lim_{x \to 0} \frac{\cos(x) - 1}{x}$.
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\end{proof}
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\subsection{Die Zahl $\pi$}
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Ziel: Analytische Definition von $\pi \in \R$.
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\begin{satz}[und Definition]
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Die Funktion $\cos\colon [0,2] \to \R$ hat genau eine Nullstelle
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im Intervall $[0,2]$, welche mit $\frac{\pi}{2}$ bezeichnet
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wird ($\pi := 2 \frac{\pi}{2}$ ).
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\end{satz}
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\begin{proof}
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in 4 Schritten.
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Schritt 1 / Lemma 1: $\cos(2) \le -\frac{1}{3}$. \\
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Restgliedabschätzung liefert ($|x| \le 5$ ).
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\[
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\cos(x) = 1 - \frac{x^2}{2} + R_4(x) \text{ mit } |R_4(x)| \le \frac{|x|^{4}}{24}
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.\] $\implies$
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\[
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\cos(2) = 1 - 2 + \underbrace{R_4(2)}_{\le \frac{16}{24} = \frac{2}{3}} \le -1 + \frac{2}{3} = -\frac{1}{3}
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.\]
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Schritt 2 / Lemma 2: $\sin(x) > 0$ $\forall x \in \; ]0, 2[$\\
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Es gilt
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\begin{align*}
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\sin(x) = x + R_3(x) = x (1 + \frac{R_3(x)}{x})
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\left| \frac{R_3(x)}{x} \right| \le \frac{|x|^2}{6}
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\stackrel{0 < x \le 2}{\le} \frac{4}{6} = \frac{2}{3}
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\intertext{$\implies$}
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1 + \frac{R_3(x)}{x} \ge \frac{1}{3}
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.\end{align*}
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Schritt 3 / Lemma 3: $\cos: [0,2] \to \R$ ist streng monoton fallend.\\
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Sei $0 \le y < x \le 2$. Dann gilt
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\begin{align*}
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\cos(x) - \cos(y) \stackrel{\text{Additionstheorem}}{=}
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- 2 \underbrace{\sin\left( \frac{x+y}{2} \right)}_{> 0}
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\underbrace{\sin\left( \frac{x-y}{2} \right)}_{> 0} < 0
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.\end{align*}
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Schritt 4 (Beweis der Definition von $\pi$ )
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$\cos(0) = 1$ (nach Definition).
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\[
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\cos(2) \le - \frac{1}{3} \stackrel{\text{Zwischenwertsatz}}{\implies}
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\exists x_0 \in [0,2] \text{ mit } \cos(x_0) = 0
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.\] Nach Lemma 3 ist $x_0$ eindeutig.
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\end{proof}
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\begin{korrolar}[Spezielle Werte von $\exp$]
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Es gilt: $e^{i \frac{\pi}{2}} = i$,
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$e^{i \pi} = -1$, $e^{i \frac{3\pi}{2}} = -i$, $e^{2\pi i} = 1$
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\end{korrolar}
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\begin{proof}
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Übung.
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\end{proof}
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\begin{korrolar}[Eigenschaften Sinus / Cosinus]
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$\forall x \in \R$ gilt:
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\begin{enumerate}[(i)]
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\item $\cos(x + 2\pi) = \cos(x) \quad \sin(x+2\pi) = \sin(x)$ \\
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$2 \pi$: Periodizität
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\item $\cos(x + \pi) = - \cos(x) \quad \sin(x+ \pi) = - \sin(x)$
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\item $\cos(x) = \sin(\frac{\pi}{2} - x) \quad \sin(x) = \cos(\frac{\pi}{2} - x)$
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\item Nullstellen von $\sin / \cos$.\\
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$\{x \in \R | \sin x = 0\} = \{x = k\pi | k \in \Z\} $ \\
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$\{x \in \R | \cos x = 0\} = \{x = \left(k+\frac{1}{2}\right)\pi | k \in \Z\} $ \\
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\end{enumerate}
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\end{korrolar}
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\begin{proof}
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folgt aus den Additionstheoremen, der Definition von $\frac{\pi}{2}$,
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den speziellen Werten von $\exp$ und folgender Tabelle
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\begin{tabular}{l|l|l|l|l|l}
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x & 0 & $\frac{\pi}{2}$ & $\pi$ & $\frac{3}{2} \pi$ & $2 \pi$ \\ \hline
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$\cos x$ & 1 & 0 & $-1$ & 0 & 1 \\ \hline
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$\sin x$ & 0 & 1 & 0 & $-1$ & 0 \\
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\end{tabular}.
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\end{proof}
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\begin{korrolar}[$e^{z} = 1$]
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Es gilt $\{z \in \mathbb{C} | e^{z} = 1\} = \{i 2 \pi k | k \in \Z\} $
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\end{korrolar}
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\begin{proof}
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ohne Beweis.
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\end{proof}
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\begin{definition}[Tangens, Cotangens]
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\begin{enumerate}[(i)]
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\item Die Tangensfunktion
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\begin{align*}
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&\tan: \R \setminus \{x = (k + \frac{1}{2}) \pi | k \in \Z\}
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\to \R
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\intertext{ist definiert durch}
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&\tan x := \frac{\sin x}{\cos x}
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.\end{align*}
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\item Die Cotangensfunktion
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\begin{align*}
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&\cot: \R \setminus \{x = k \pi | k \in \Z\} \to \R
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\intertext{ist definiert durch}
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&\cot x := \frac{\cos(x)}{\sin(x)}
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.\end{align*}
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\end{enumerate}
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\end{definition}
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\begin{figure}[htpb]
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\centering
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\begin{tikzpicture}
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\begin{axis}%
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[grid=both,
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minor tick num=4,
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grid style={line width=.1pt, draw=gray!10},
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major grid style={line width=.2pt,draw=gray!50},
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axis lines=middle,
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enlargelimits={abs=0.2},
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ymax=5,
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ymin=-5
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]
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\addplot[domain=-3:3,samples=50,smooth,red] {tan(deg(x))};
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\end{axis}
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\end{tikzpicture}
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\caption{$\tan(x)$}
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\end{figure}
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\begin{figure}[htpb]
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\centering
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\begin{tikzpicture}
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\begin{axis}%
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[grid=both,
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minor tick num=4,
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grid style={line width=.1pt, draw=gray!10},
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major grid style={line width=.2pt,draw=gray!50},
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axis lines=middle,
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enlargelimits={abs=0.2},
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ymax=5,
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ymin=-5
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]
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\addplot[domain=-3:3,samples=50,smooth,red] {cot(deg(x))};
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\end{axis}
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\end{tikzpicture}
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\caption{$\cot(x)$}
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\end{figure}
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\begin{definition}[Arcusfunktionen (Umkehrfunktionen der Trigonometrischen
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Funktionen)]
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\begin{enumerate}[(i)]
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\item $\cos\colon [0, \pi] \to [-1, 1]$ ist streng monoton fallend
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und bijektiv. Die Umkehrfunktion heißt
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Arcus-Cosinus.
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\[
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\arccos: [-1,1] \to [0, \pi]
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.\]
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\item $\sin\colon \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \to [-1, 1]$
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ist streng monoton wachsend und bijektiv. Die Umkehrfunktion
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heißt Arcus-Sinus.
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\[
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\arcsin: [-1,1] \to \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]
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.\]
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\item $\tan\colon \; ] - \frac{\pi}{2}, \frac{\pi}{2} [ \to \R$ ist streng
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monoton wachsend und bijektiv. Die Umkehrfunktion heißt
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Arcus-Tangens.
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\[
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\arctan: \R \to ] - \frac{\pi}{2}, \frac{\pi}{2} [
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.\]
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\end{enumerate}
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\end{definition}
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\begin{satz}[Polarkoordinaten]
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Jedes $z \in \mathbb{C}$ lässt sich schreiben als
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$z = r\cdot e^{i \varphi}$, $\varphi \in \R$ und
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$r = |z| \in [0, \infty[$.
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Für $z \neq 0$ ist $\varphi$ bis auf ein ganzzahliges Vielfaches von
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$2\pi$ eindeutig bestimmt.
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\end{satz}
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\begin{proof}
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Rannacher.
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\end{proof}
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\end{document}
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