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\section{Allgemeines} \begin{sectionbox} - -\subsection{Bruchrechnung}\label{bruchrechnung} -\begin{itemize} -\item - \(\frac{a}{b}\) : \(\frac{c}{d}\) = Multiplikation mit Kehrwert = - \(\frac{ab}{bd}\) -\item - Brüche kürzen: nur Faktoren, nicht Summanden! - - \begin{itemize} - \item - \(\frac{2}{2\ *\ 3}\) = \(\frac{2}{2}\) * \(\frac{1}{3}\) = - \(\frac{1}{3}\) - \end{itemize} -\item - Potenzen siehe „Expotentialfunktion`` -\end{itemize} - \subsection{Zahlenmengen}\label{zahlenmengen} \begin{itemize} @@ -73,35 +55,71 @@ \subsection{Binomische Formeln}\label{binomische-formeln} -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - ${(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2}$ -\item - ${(a -- b)}^{2} = {a}^{2} - 2ab + {b}^{2}$ -\item - $(a + b) (a -- b) = {a}^{2} - {b}^{2}$ -\end{enumerate} +\begin{tablebox}{ll} +1. Binomische Formel: & ${\left(a + b \right)}^{2} = {a}^{2} + 2ab + {b}^{2}$ \\ +2. Binomische Formel: & ${\left(a - b \right)}^{2} = {a}^{2} - 2ab + {b}^{2}$ \\ +3. Binomische Formel: & $\left(a + b \right) \left(a - b \right) = {a}^{2} - {b}^{2}$ \\ +Bnomischer Lehrsatz: & ${\left( a + b \right)}^{n} = \sum _{ k = 0 }^{ n }{ { a }^{ n-k }{ b }^{ k } } $ \\ +\end{tablebox} + +Den Binomischen Lehrsatz kannst du auch aus dem pascalschen Dreieck entnehmen. -\subsection{Binomischer Lehrsatz}\label{binomischer-lehrsatz} +\subsection{Sinus \& Cosinus} +\begin{tablebox}{c|c|c|c|c} + Bogenmaß & Grad & $\sin{x}$ & $\cos{x}$ & $\tan{x}$ \\ \hline + $0 \pi$ & $0^\circ$ & $0$ & $1$ & $0$\\ \hline + $\cfrac{1}{6} \pi$ & $30^\circ$ & $\cfrac{1}{2}$ & $\cfrac{1}{2}\sqrt{3}$ & $\cfrac{1}{\sqrt{3}}$\\ \hline + $\cfrac{1}{4} \pi$ & $45^\circ$ & $\cfrac{1}{2}\sqrt{2}$ & $\cfrac{1}{2}\sqrt{2}$ & $1$\\ \hline + $\cfrac{1}{3} \pi$ & $60^\circ$ & $\cfrac{1}{2}\sqrt{3}$ & $\cfrac{1}{2}$ & $\sqrt{3}$\\ \hline + $\cfrac{1}{2} \pi$ & $90^\circ$ & $1$ & $0$ & $\pm \infty$\\ \hline + $\cfrac{2}{3} \pi$ & $120^\circ$ & $\cfrac{1}{2}\sqrt{3}$ & $-\cfrac{1}{2}$ & $-\sqrt{3}$\\ \hline + $\cfrac{3}{4} \pi$ & $135^\circ$ & $\cfrac{1}{2}\sqrt{2}$ & $-\cfrac{1}{2}\sqrt{2}$ & $-1$\\ \hline + $\cfrac{5}{6} \pi$ & $150^\circ$ & $\cfrac{1}{2}$ & $-\cfrac{1}{2}\sqrt{3}$ & $-\cfrac{1}{\sqrt{3}}$\\ \hline + $\cfrac{1}{1} \pi$ & $180^\circ$ & $0$ & $-1$ & $0$\\ \hline + $\cfrac{7}{6} \pi$ & $210^\circ$ & $-\cfrac{1}{2}$ & $-\cfrac{1}{2}\sqrt{3}$ & $\cfrac{1}{\sqrt{3}}$\\ \hline + $\cfrac{5}{4} \pi$ & $225^\circ$ & $-\cfrac{1}{2}\sqrt{2}$ & $-\cfrac{1}{2}\sqrt{2}$ & $1$\\ \hline + $\cfrac{4}{3} \pi$ & $240^\circ$ & $-\cfrac{1}{2}\sqrt{3}$ & $-\cfrac{1}{2}$ & $\sqrt{3}$\\ \hline + $\cfrac{3}{2} \pi$ & $270^\circ$ & $-1$ & $0$ & $\pm \infty$\\ \hline + $\cfrac{5}{3} \pi$ & $300^\circ$ & $-\cfrac{1}{2}\sqrt{3}$ & $\cfrac{1}{2}$ & $-\sqrt{3}$\\ \hline + $\cfrac{7}{4} \pi$ & $315^\circ$ & $-\cfrac{1}{2}\sqrt{2}$ & $\cfrac{1}{2}\sqrt{2}$ & $-1$\\ \hline + $\cfrac{11}{6} \pi$ & $330^\circ$ & $-\cfrac{1}{2}$ & $\cfrac{1}{2}\sqrt{3}$ & $-\cfrac{1}{\sqrt{3}}$\\ \hline +\end{tablebox} + +$\cfrac{1}{2}\sqrt{2} \cong 0.70710678$ und $\cfrac{1}{2}\sqrt{3} \cong 0.8660254$ \\ sowie $\cfrac{1}{\sqrt{3}} \cong 0577350269$ +\subsection{Bruchrechnung}\label{bruchrechnung} \begin{itemize} \item - \({(a + b)}^{n}\) = \(\sum_{k = 0}^{n}{a^{n - k}b^{k}}\) + \(\frac{a}{b}\) : \(\frac{c}{d}\) = Multiplikation mit Kehrwert = + \(\frac{ab}{bd}\) \item - z.b.: 1 \(\bullet\) \(a^{5} \bullet\) \(b^{0}\) + \ldots{} -\end{itemize} + Brüche kürzen: nur Faktoren, nicht Summanden! -\subsection{Sinus \& Cosinus} -\begin{tablebox}{l|l} - $\cos 0^\circ = 1$ & $\sin 0^\circ = 0$ \\ - $\cos 30^\circ = \cfrac{1}{2}\sqrt{3} \cong 0.8660254$ & $\sin 30^\circ = \cfrac{1}{2}$ \\ - $\cos 45^\circ = \cfrac{1}{2}\sqrt{2} \cong 0.70710678$ & $\sin 45^\circ = \cfrac{1}{\sqrt{2}} \cong 0.70710678$ \\ - $\cos 90^\circ = 0$ & $\sin 90^\circ = 1$ \\ - \end{tablebox} + \begin{itemize} + \item + \(\frac{2}{2\ *\ 3}\) = \(\frac{2}{2}\) * \(\frac{1}{3}\) = + \(\frac{1}{3}\) + \end{itemize} +\end{itemize} \end{sectionbox} +% Weiterführend Allgemeines +% ---------------------------------------------------------------------- +\begin{sectionbox} + +\subsection{Potenzrechnung} +\begin{tablebox}{ll} +${a}^{n} \cdot {a}^{m} = {a}^{n+m}$ & $\frac{ {a}^{n} }{ {a}^{m} } = {a}^{n-m}$ \\ +${\left({a}^{m}\right)}^{n} = {\left({a}^{n}\right)}^{m} = {a}^{m \cdot n}$ & ${a}^{n} \cdot {b}^{n} = {\left(a \cdot b \right)}^{n}$ \\ +$\frac{{a}^{n}}{{b}^{n}} = {\left(\cfrac{a}{b}\right)}^{n}$ & ${-a}^{-1} = \cfrac{-1}{a} = \cfrac{{a}^{-1}}{-1}$\\ +${e}^{lnx} = x$ & ${a}^{-n} = \frac{1}{ {a}^{n} } $ +\end{tablebox} + +\subsection{Wurzelrechnung} + + +\end{sectionbox} % Mengenlehre % ----------------------------------------------------------------------