trigonometrie-allgemeines-dreieck

Formeln

kosinussatz $a=\sqrt{b^2+c^2-2\ast b\ast c-\cos (\alpha )}$ auf alpha umgeformnt $\alpha =\arccos (\frac{-a^2+b^2+c^2}{2bc})$ sinussatz $\frac{a}{\sin (\alpha )}=\frac{b}{\sin (\beta )}=\frac{c}{\sin (\gamma )}$ auf winkel umgeformnt $\gamma =\arcsin (\frac{c\ast \sin (\alpha )}{a})$ $\alpha =\arcsin (\frac{a\ast \sin (\alpha )}{c})$ Flächenihalt $A=\frac{1}{2}\ast b\ast c\ast \sin (\alpha )=\frac{1}{2}\ast a\ast c\ast \sin (\beta )=\frac{1}{2}\ast a\ast b\ast \sin (\gamma )$ paralelogramm fläche $A=a\ast b\ast \sin (\alpha )$

txtbsp

$$||"id":1002272221953||\alpha =85°\beta =42°c=17km\gamma =180-85-42=53°\frac{a}{\sin (85)}=\frac{17}{\sin (53)}|\ast \sin (85)a=\frac{17}{\sin (53)}\ast \sin (85)=21.2b=\sqrt{21.2^2+17^2-2\ast 21.2\ast 17\ast \cos (42)}=14.24km$$

$$||"id":357667955206||\frac{a}{\sin (8.5)}=\frac{2}{\sin (4.5)}|\ast \sin (8.5)a=\frac{2}{\sin (4.5)}\ast \sin (8.5)=3.77\cos (13°)=\frac{3.77}{a}|\ast a\cos (13°)\ast a=3.77|:\cos (13)a=\frac{3.77}{\cos (13)}=3.87x=3.87-0.03=8.84$$

$$||"id":131987418342||d=2.5m\alpha =35°\sin (35)=\frac{x}{40}|\ast 40x=\sin (35)\ast 40=22.94\tan (35)=\frac{22.94}{z}|\ast z\tan (35)\ast z=22.94|:\tan (35)z=\frac{22.94}{\tan (35)}=32.76zz=32.76+1.25=34.01\text{ohm}=180-90-35=55zzz=\sqrt{34.01^2+22.94^2}=41.023zzzz=41.023-1.25=39.77BD=40-39.77=0.23$$