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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: 01
$\begin{array}{l}
\text{Gegeben:}\\\text{Länge der Seite} \qquad c \qquad [m] \\
\text{Länge der Seite} \qquad b \qquad [m] \\
\text{Länge der Seite} \qquad a \qquad [m] \\
\\ \text{Gesucht:} \\\text{Winkel} \qquad \alpha \qquad [^{\circ}] \\
\\ cos\alpha = \frac{b^{2} +c^{2} -a^{2} }{2\cdot b\cdot c}\\ \textbf{Gegeben:} \\ c=3m \qquad b=7m \qquad a=9m \qquad \\ \\ \textbf{Rechnung:} \\
cos\alpha = \frac{b^{2} +c^{2} -a^{2} }{2\cdot b\cdot c} \\
c=3m\\
b=7m\\
a=9m\\
cos\alpha = \frac{(7m)^{2} +(3m)^{2} -(9m)^{2} }{2\cdot 7m\cdot 3m}\\\\\alpha=123^{\circ}
\\\\\\ \small \begin{array}{|l|} \hline c=\\ \hline 3 m \\ \hline 30 dm \\ \hline 300 cm \\ \hline 3\cdot 10^{3} mm \\ \hline 3\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline b=\\ \hline 7 m \\ \hline 70 dm \\ \hline 700 cm \\ \hline 7\cdot 10^{3} mm \\ \hline 7\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline a=\\ \hline 9 m \\ \hline 90 dm \\ \hline 900 cm \\ \hline 9\cdot 10^{3} mm \\ \hline 9\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline alpha=\\ \hline 123 ° \\ \hline 7,39\cdot 10^{3} \text{'} \\ \hline 4,44\cdot 10^{5} \text{''} \\ \hline 137 gon \\ \hline 2,15 rad \\ \hline \end{array} \end{array}$