Geometrie-Trigonometrie-Sinussatz


  • $ \frac{a}{ sin\alpha} = \frac{b}{ sin\beta }= \frac{c}{ sin\gamma }$
    $a = \frac{b\cdot sin\alpha }{ sin\beta }$
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    $sin\alpha = \frac{a\cdot sin\beta }{ b}$
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Beispiel Nr: 07
$ \text{Gegeben:}\\\text{Winkel} \qquad \beta \qquad [^{\circ}] \\ \text{Winkel} \qquad \alpha \qquad [^{\circ}] \\ \text{Länge der Seite} \qquad b \qquad [m] \\ \\ \text{Gesucht:} \\\text{Länge der Seite} \qquad a \qquad [m] \\ \\ a = \frac{b\cdot sin\alpha }{ sin\beta }\\ \textbf{Gegeben:} \\ \beta=20^{\circ} \qquad \alpha=50^{\circ} \qquad b=\frac{2}{5}m \qquad \\ \\ \textbf{Rechnung:} \\ a = \frac{b\cdot sin\alpha }{ sin\beta } \\ \beta=20^{\circ}\\ \alpha=50^{\circ}\\ b=\frac{2}{5}m\\ a = \frac{\frac{2}{5}m\cdot sin50^{\circ} }{ sin20^{\circ} }\\\\a=0,896m \\\\\\ \small \begin{array}{|l|} \hline beta=\\ \hline 20 ° \\ \hline 1,2\cdot 10^{3} \text{'} \\ \hline 7,2\cdot 10^{4} \text{''} \\ \hline 22\frac{2}{9} gon \\ \hline 0,349 rad \\ \hline \end{array} \small \begin{array}{|l|} \hline alpha=\\ \hline 50 ° \\ \hline 3\cdot 10^{3} \text{'} \\ \hline 1,8\cdot 10^{5} \text{''} \\ \hline 55\frac{5}{9} gon \\ \hline 0,873 rad \\ \hline \end{array} \small \begin{array}{|l|} \hline b=\\ \hline \frac{2}{5} m \\ \hline 4 dm \\ \hline 40 cm \\ \hline 400 mm \\ \hline 4\cdot 10^{5} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline a=\\ \hline 0,896 m \\ \hline 8,96 dm \\ \hline 89,6 cm \\ \hline 896 mm \\ \hline 8,96\cdot 10^{5} \mu m \\ \hline \end{array}$