Erweiterung der Sinus/Cosinus Tabelle

Formelsammlung.tex

@@ -37,24 +37,6 @@
 \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}
@@ -73,36 +55,72 @@
 
 \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
+  Brüche kürzen: nur Faktoren, nicht Summanden!
+
+  \begin{itemize}
   \item
-  z.b.: 1 \(\bullet\) \(a^{5} \bullet\) \(b^{0}\) + \ldots{}
+    \(\frac{2}{2\ *\ 3}\) = \(\frac{2}{2}\) * \(\frac{1}{3}\) =
+    \(\frac{1}{3}\)
+  \end{itemize}
 \end{itemize}
 
-\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{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}
 
-\end{sectionbox}
+\subsection{Wurzelrechnung}
 
 
+\end{sectionbox}
+
 % Mengenlehre
 % ----------------------------------------------------------------------
 \section{Mengenlehre}