Berechne den Winkel $\alpha \backslash large\backslash alpha$

Du kennst die Seite $a=7cma=7\backslash \; cm$ und die Seite $\overline{SB}=\frac{\overline{DB}}{2}=5cm\backslash overline\{SB\}=\backslash frac\{\backslash overline\{DB\}\}\{2\}=5\backslash \; cm$; $\sin {\displaystyle \frac{\alpha }{2}}={\displaystyle \frac{\overline{SB}}{a}};\sin {\displaystyle \frac{\alpha }{2}}={\displaystyle \frac{5cm}{7cm}};\sin {\displaystyle \frac{\alpha }{2}}=\mathrm{0,7143}\Rightarrow \alpha ={\mathrm{91,18}}^{\circ }\backslash sin\backslash dfrac\{\backslash alpha\}\{2\}=\backslash dfrac\{\backslash overline\{SB\}\}\{a\};\backslash quad\; \backslash sin\backslash dfrac\{\backslash alpha\}\{2\}=\backslash dfrac\{5\backslash \; cm\}\{7\backslash \; cm\};\backslash quad\; \backslash sin\backslash dfrac\{\backslash alpha\}\{2\}=0\{,\}7143\backslash Rightarrow\backslash \; \backslash alpha=91\{,\}18^\{\backslash circ\}$

Berechne den Winkel $\gamma \backslash large\backslash gamma$

Du kennst die Seite $b=c=6cmb=c=6\backslash \; cm$ und die Seite $\overline{SB}={\displaystyle \frac{\overline{DB}}{2}}=5cm\backslash overline\{SB\}=\backslash dfrac\{\backslash overline\{DB\}\}\{2\}=5\backslash \; cm$ $\sin {\displaystyle \frac{\gamma }{2}}={\displaystyle \frac{\overline{SB}}{b}};\sin {\displaystyle \frac{\gamma }{2}}={\displaystyle \frac{5cm}{6cm}};sin{\displaystyle \frac{\gamma }{2}}=\mathrm{0,8333}\Rightarrow \gamma ={\mathrm{112,88}}^{\circ }\backslash sin\backslash dfrac\{\backslash gamma\}\{2\}=\backslash dfrac\{\backslash overline\{SB\}\}\{b\};\backslash quad\; \backslash sin\backslash dfrac\{\backslash gamma\}\{2\}=\backslash dfrac\{5\backslash \; cm\}\{6\backslash \; cm\};\backslash quad\; sin\backslash dfrac\{\backslash gamma\}\{2\}=\; 0\{,\}8333\backslash Rightarrow\backslash \; \backslash gamma=\; 112\{,\}88^\backslash circ$

Berechne den Winkel $\beta \backslash beta$

Die Winkelsumme im Viereck beträgt ${360}^{\circ }360^\backslash circ$, also gilt: ${360}^{\circ }=\alpha +2\cdot \beta +\gamma \Rightarrow \beta ={\displaystyle \frac{{360}^{\circ }-\alpha -\gamma }{2}}={\displaystyle \frac{{360}^{\circ }-{\mathrm{91,18}}^{\circ }-{\mathrm{112,88}}^{\circ }}{2}}\beta ={\mathrm{77,97}}^{\circ }\backslash \backslash 360^\backslash circ=\; \backslash alpha+2\backslash cdot\backslash beta+\backslash gamma\; \backslash Rightarrow\backslash beta=\backslash dfrac\{360^\backslash circ-\backslash alpha-\backslash gamma\}\{2\}=\backslash dfrac\{360^\backslash circ-91\{,\}18^\backslash circ-112\{,\}88^\backslash circ\}\{2\}\backslash \backslash \; \backslash beta=77\{,\}97^\backslash circ$