Weitere Formeln

Die folgenden Formeln folgen nach längeren Termumformungen aus $\alpha + \beta + \gamma = 180^\circ$, gelten also allgemein für drei beliebige Winkel $\alpha$, $\beta$ und $\gamma$ mit der Eigenschaft $\alpha + \beta + \gamma = 180^\circ$, solange die in den Formeln vorkommenden Funktionen wohldefiniert sind (letzteres betrifft nur die Formeln, in denen Tangens und Kotangens vorkommen).

$$\tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \cdot \tan \beta \cdot \tan \gamma \,$$

$$\cot \beta \cdot \cot \gamma + \cot \gamma \cdot \cot \alpha + \cot \alpha \cdot \cot \beta = 1$$

$$\cot \frac{\alpha }{2}+ \cot \frac{\beta }{2}+ \cot \frac{\gamma }{2} = \cot \frac{\alpha }{2} \cdot \cot \frac {\beta }{2} \cdot \cot \frac{\gamma }{2}$$

$$\tan \frac{\beta }{2}\tan \frac{\gamma }{2}+\tan \frac{\gamma }{2}\tan \frac{\alpha }{2}+\tan \frac{\alpha }{2}\tan \frac{\beta }{2} = 1$$

$$\sin \alpha +\sin \beta +\sin \gamma = 4\cos \frac{\alpha }{2}\cos \frac{\beta }{2}\cos \frac{\gamma }{2}$$

$$-\sin \alpha +\sin \beta +\sin \gamma = 4\cos \frac{\alpha }{2}\sin \frac{\beta }{2}\sin \frac{\gamma }{2}$$

$$\cos \alpha +\cos \beta +\cos \gamma = 4\sin \frac{\alpha }{2}\sin \frac{\beta }{2}\sin \frac{\gamma }{2}+1$$

$$-\cos \alpha +\cos \beta +\cos \gamma = 4\sin \frac{\alpha }{2}\cos \frac{\beta }{2}\cos \frac{\gamma }{2}-1$$

$$\sin (2\alpha) +\sin (2\beta) +\sin (2\gamma) = 4\sin \alpha \sin \beta \sin \gamma \,$$

$$-\sin (2\alpha) +\sin (2\beta) +\sin (2\gamma) = 4\sin \alpha \cos \beta \cos \gamma \,$$

$$\cos (2\alpha) +\cos (2\beta) +\cos (2\gamma) = -4\cos \alpha \cos \beta \cos \gamma -1 \,$$

$$-\cos (2\alpha) +\cos (2\beta) +\cos (2\gamma) = -4\cos \alpha \sin \beta \sin \gamma +1 \,$$

$$\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma = 2 \cos \alpha \cos \beta \cos \gamma +2 \,$$

$$-\sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma = 2 \cos \alpha \sin \beta \sin \gamma \,$$

$$\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma = -2 \cos \alpha \cos \beta \cos \gamma +1 \,$$

$$-\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma = -2 \cos \alpha \sin \beta \sin \gamma +1 \,$$

$$-\sin ^{2} (2\alpha) +\sin ^{2} (2\beta) +\sin ^{2} (2\gamma) = -2\cos (2\alpha) \,\sin (2\beta) \,\sin (2\gamma)$$

$$-\cos ^{2} (2\alpha) +\cos ^{2} (2\beta) +\cos ^{2} (2\gamma) = 2\cos (2\alpha) \,\sin (2\beta) \,\sin (2\gamma) +1$$