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anhang.tex
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anhang.tex
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\label{LastPage}
\section{Wichtige Formeln}
\subsection{Funktionswerte für Winkelargumente}
\renewcommand{\arraystretch}{1.5}
\begin{minipage}{5cm}
\begin{tabular}[c]{ |c|c||c|c|c| }
\hline
deg & rad & sin & cos & tan\\
\hline
0\symbol{23} & 0 & 0 & 1 & 0\\
\hline
30\symbol{23} & $\frac{\pi}{6}$ & $\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$ &
$\frac{\sqrt{3}}{3}$\\
\hline
45\symbol{23} & $\frac{\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$
& 1\\
\hline
60\symbol{23} & $\frac{\pi}{3}$ & $\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$ &
$\sqrt{3}$\\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{4.3cm}
\begin{tabular}[c]{ |c|c||c|c|}
\hline
deg & rad & sin & cos\\
\hline
90\symbol{23} & $\frac{\pi}{2}$ & 1 & 0\\
\hline
120\symbol{23} & $\frac{2\pi}{3}$ & $\frac{\sqrt{3}}{2}$ & $-\frac{1}{2}$ \\
\hline
135\symbol{23} & $\frac{3\pi}{4}$ & $\frac{\sqrt{2}}{2}$ & $-\frac{\sqrt{2}}{2}$\\
\hline
150\symbol{23} & $\frac{5\pi}{6}$ & $\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$\\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{4.5cm}
\begin{tabular}[c]{ |c|c||c|c| }
\hline
deg & rad & sin & cos\\
\hline
180\symbol{23} & $\pi$ & 0 & -1\\
\hline
210\symbol{23} & $\frac{7\pi}{6}$ & $-\frac{1}{2}$ & $-\frac{\sqrt{3}}{2}$\\
\hline
225\symbol{23} & $\frac{5\pi}{4}$ & $-\frac{\sqrt{2}}{2}$ & $-\frac{\sqrt{2}}{2}$\\
\hline
240\symbol{23} & $\frac{4\pi}{3}$ & $-\frac{\sqrt{3}}{2}$ & $-\frac{1}{2}$\\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{4.5cm}
\begin{tabular}[c]{ |c|c||c|c| }
\hline
deg & rad & sin & cos\\
\hline
270\symbol{23} & $\frac{3\pi}{2}$ & -1 & 0\\
\hline
300\symbol{23} & $\frac{5\pi}{3}$ & $-\frac{\sqrt{3}}{2}$ & $\frac{1}{2}$\\
\hline
315\symbol{23} & $\frac{7\pi}{4}$ & $-\frac{\sqrt{2}}{2}$ & $\frac{\sqrt{2}}{2}$\\
\hline
330\symbol{23} & $\frac{11\pi}{6}$ & $-\frac{1}{2}$ & $\frac{\sqrt{3}}{2}$\\
\hline
\end{tabular}
\end{minipage}
\renewcommand{\arraystretch}{1}
$\tan^{-1}$: wenn x$>$1 dann gilt: $\tan^{-1}(x)=\frac{\Pi}{2}-\tan^{-1}(\frac{1}{x})$
\begin{minipage}{0.4\linewidth}
\includegraphics[width=\linewidth]{./bilder/winkelfunktionen.png}
\end{minipage}%
\hspace{0.01\linewidth}
\begin{minipage}{0.2\linewidth}
\begin{tabular}[c]{ |c|c|c| }
\hline
arctan(x) & deg & rad \\
\hline
2-$\sqrt{3}$ & 0\symbol{23} & 0\\
\hline
$\sqrt{2}-1$ & 22.5\symbol{23} & $\frac{\pi}{8}$\\
\hline
$\frac{1}{\sqrt{3}}$ & 30\symbol{23} & $\frac{\pi}{6}$\\
\hline
1 & 45\symbol{23} & $\frac{\pi}{4}$\\
\hline
\end{tabular}
\end{minipage}
\hspace{0.01\linewidth}
\begin{minipage}{0.3\linewidth}
\includegraphics[width=\linewidth]{./bilder/e_kurven.png}
\end{minipage}
\begin{minipage}[t]{9cm}
\subsection{Additionstheoreme}
$\sin(a \pm b)=\sin(a) \cdot \cos(b) \pm \cos(a) \cdot \sin(b)$\\
$\cos(a \pm b)=\cos(a) \cdot \cos(b) \mp \sin(a) \cdot \sin(b)$\\
$\tan(a \pm b)=\frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \cdot \tan(b)}$
\subsection{Doppel- und Halbwinkel}
$\sin(2a)=2\sin(a)\cos(a)$\\
$\cos(2a)=\cos^2(a)-\sin^2(a)=2\cos^2(a)-1=1-2\sin^2(a)$\\
$\cos^2 \left(\frac{a}{2}\right)=\frac{1+\cos(a)}{2} \qquad
\sin^2 \left(\frac{a}{2}\right)=\frac{1-\cos(a)}{2}$
\subsection{Produkte}
$\sin(a)\sin(b)=\frac{1}{2}(\cos(a-b)-cos(a+b))$\\
$\cos(a)\cos(b)=\frac{1}{2}(\cos(a-b)+cos(a+b))$\\
$\sin(a)\cos(b)=\frac{1}{2}(\sin(a-b)+\sin(a+b))$
\end{minipage}
\begin{minipage}[t]{6cm}
\subsection{Quadrantenbeziehungen}
\begin{tabbing}
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \= \kill
$\sin(-a)=-\sin(a)$ \> $\cos(-a)=\cos(a)$\\
$\sin(\pi - a)=\sin(a)$ \> $\cos(\pi - a)=-\cos(a)$\\
$\sin(\pi + a)=-\sin(a)$ \> $\cos(\pi +a)=-\cos(a)$\\
$\sin\left(\frac{\pi}{2}-a \right)=\sin\left(\frac{\pi}{2}+a \right)=\cos(a)$\\
$\cos\left(\frac{\pi}{2}-a \right)=-\cos\left(\frac{\pi}{2}+a \right)=\sin(a)$
\end{tabbing}
\subsection{Summe und Differenz}
$\sin(a)+\sin(b)=2 \cdot \sin \left(\frac{a+b}{2}\right) \cdot
\cos\left(\frac{a-b}{2}\right)$\\
$\sin(a)-\sin(b)=2 \cdot \sin \left(\frac{a-b}{2}\right) \cdot
\cos\left(\frac{a+b}{2}\right)$\\
$\cos(a)+\cos(b)=2 \cdot \cos \left(\frac{a+b}{2}\right) \cdot
\cos\left(\frac{a-b}{2}\right)$\\
$\cos(a)-\cos(b)=-2 \cdot \sin \left(\frac{a+b}{2}\right) \cdot
\cos\left(\frac{a-b}{2}\right)$\\
$\tan(a) \pm \tan(b)=\frac{\sin(a \pm b)}{\cos(a)\cos(b)}$
\end{minipage}
%\subsection{Skalarprodukt}
% \begin{tabular}{|p{3cm}|p{7cm}|p{7cm}|}
% \hline
% & Reelle Fourierreihe & Komplexe Fourierreihe \\
% \hline
% Skalarprodukt & $\langle f;g \rangle = \frac{2}{T}\int\limits_0^Tf(t)\cdot
% g(t)\cdot dt $ & $\langle f;g \rangle = \frac{1}{T}\int\limits_0^Tf(t)\cdot
% \overline{g(t)}\cdot dt $ \\
% & $(I)\ \langle f;f \rangle > 0\ f"ur\ f\neq 0$ &
% $(I)\ \langle f;f \rangle > 0\ f"ur\ f\neq 0$ \\
% & $(II)\ \langle f;g \rangle = \langle g;f \rangle $ &
% $(II)\ \langle f;g \rangle = \overline{\langle g;f \rangle} $ \\
% & $(III) \langle r\cdot f;g\rangle = r\cdot\langle f;g\rangle,\ r\epsilon
% \mathbb R$ & $(III) \langle c\cdot f;g\rangle = c\cdot\langle f;g\rangle,\
% c\epsilon \mathbb R$
% \\
% \hline
% Länge (Norm): & \multicolumn{2}{|l|}{
% $\parallel f \parallel = \sqrt{\langle
% f;f \rangle} \Rightarrow y=3x^2+4 \rightarrow \sqrt{3\cdot3+4\cdot4}=5$
% } \\
% & \multicolumn{2}{|l|}{ $ (I)\ \parallel f \parallel > 0 f"ur f \neq 0$} \\
% & \multicolumn{2}{|l|}{ $ (II)\ \parallel f+g \parallel \leq \parallel f
% \parallel + \parallel g \parallel (Dreiecksgleichung)$} \\
% & \multicolumn{2}{|l|}{ $ (III)\ \parallel r\cdot f \parallel = |r| \cdot
% \parallel f \parallel , \ r \epsilon \mathbb R bzw. \parallel c\cdot
% f\parallel = |c|\cdot \parallel f \parallel, c\epsilon\mathbb C$} \\
% \hline
% Abstand: & \multicolumn{2}{|l|}{ $ \parallel f-g \parallel =
% |\sqrt{\langle f-g;f-g \rangle}| \Rightarrow
% |p-q|=(3x^2+4)-(x^2+7)=2x^2+11 \rightarrow
% \sqrt{2\cdot2+11\cdot11}=5\cdot\sqrt{5}$}
% \\
% \hline
% Winkel, Orthogonalit"at &
% \multicolumn{2}{|l|}{
% $\sphericalangle(f;g)=\arccos\left( \frac{\langle f;g \rangle}{\parallel f
% \parallel \cdot \parallel g \parallel} \right) = \arccos\left(\frac{\langle f;g
% \rangle}{\sqrt{\langle f;f\rangle \cdot \langle g;g\rangle}}\right)$} \\
% & \multicolumn{2}{|l|}{$f \perp g\Leftrightarrow \langle f;g \rangle = 0$}
% \\
%
%
% \hline
% \end{tabular}
\section{Diverses}
\begin{tabbing}
xxxxxxxxxxxxxxxxxxxxxxxxxxxx \= xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx \= \kill
$f'(z) = \lim \limits_{\Delta z \rightarrow 0} \frac{f(z + \Delta z) -
f(z)}{\Delta z}$ \> $(a + b)^n = \sum_{k=0}^{n} \binom n k a^{n-k} \cdot b^k$ \>
$(a \pm b)^3 =a^3 \pm 3 a^{2} b + 3 a b^2 \pm b^3 $\\ \\
$x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ \> $\binom n k = \frac{n!}{k!
\cdot (n-k)!}$ \> $(a \pm b)^4 =a^4 \pm 4 a^{3} b + 6a^2b^2 \pm 4 a b^3 +
b^4$\\
\end{tabbing}
%\subsection{Integrale}
% Partielle Integration: $\int^{b}_{a} u(x) v'(x) dx = [u(x)v(x)]^{b}_{a} - \int^{b}_{a} u'(x) v(x) dx$
% \begin{center}
% \includegraphics[width=\linewidth]{./bilder/integral1.png}
% \end{center}
% \begin{center}
% \includegraphics[width=\linewidth]{./bilder/integral2.png}
% \end{center}