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$ a = \sqrt{b^{2} +c^{2} -2\cdot b\cdot c\cdot cos\alpha } $
$ cos\alpha = \frac{b^{2} +c^{2} -a^{2} }{2\cdot b\cdot c} $
Geometrie-Trigonometrie-Kosinussatz
$ a^2 = b^{2} +c^{2} -2\cdot b\cdot c\cdot cos\alpha $
$a = \sqrt{b^{2} +c^{2} -2\cdot b\cdot c\cdot cos\alpha }$
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$cos\alpha = \frac{b^{2} +c^{2} -a^{2} }{2\cdot b\cdot c}$
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Beispiel Nr: 06
$\begin{array}{l}
\text{Gegeben:}\\\text{Länge der Seite} \qquad c \qquad [m] \\
\text{Länge der Seite} \qquad b \qquad [m] \\
\text{Länge der Seite} \qquad a \qquad [m] \\
\\ \text{Gesucht:} \\\text{Winkel} \qquad \alpha \qquad [^{\circ}] \\
\\ cos\alpha = \frac{b^{2} +c^{2} -a^{2} }{2\cdot b\cdot c}\\ \textbf{Gegeben:} \\ c=3m \qquad b=4m \qquad a=5m \qquad \\ \\ \textbf{Rechnung:} \\
cos\alpha = \frac{b^{2} +c^{2} -a^{2} }{2\cdot b\cdot c} \\
c=3m\\
b=4m\\
a=5m\\
cos\alpha = \frac{(4m)^{2} +(3m)^{2} -(5m)^{2} }{2\cdot 4m\cdot 3m}\\\\\alpha=90^{\circ}
\\\\\\ \small \begin{array}{|l|} \hline c=\\ \hline 3 m \\ \hline 30 dm \\ \hline 300 cm \\ \hline 3\cdot 10^{3} mm \\ \hline 3\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline b=\\ \hline 4 m \\ \hline 40 dm \\ \hline 400 cm \\ \hline 4\cdot 10^{3} mm \\ \hline 4\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline a=\\ \hline 5 m \\ \hline 50 dm \\ \hline 500 cm \\ \hline 5\cdot 10^{3} mm \\ \hline 5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline alpha=\\ \hline 90 ° \\ \hline 5,4\cdot 10^{3} \text{'} \\ \hline 3,24\cdot 10^{5} \text{''} \\ \hline 100 gon \\ \hline 1,57 rad \\ \hline \end{array} \end{array}$