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Question Number 59997 by aliesam last updated on 16/May/19

$$find(a,b,c)inR$$$$cos\left(a\right)+cos\left(b\right)+cos\left(c\right)=0$$$$sin\left(a\right)+sin\left(b\right)+sin\left(c\right)=0$$

Answered by MJS last updated on 17/May/19

$$\mathrm{we}\mathrm{have}\mathrm{only}2\mathrm{equations}\mathrm{in}3\mathrm{variables}\Rightarrow$$$$\mathrm{we}\mathrm{are}\mathrm{free}\mathrm{to}\mathrm{choose}\mathrm{one}$$$$\mathrm{let}a=0$$$$1+\cos b+\cos c=0$$$$\sin b+\sin c=0$$$$\mathrm{let}\cos b=x\wedge \cos c=y\Rightarrow \sin b=\pm \sqrt{1-x^2}\wedge \sin c=\pm \sqrt{1-y^2}$$$$1+x+y=0\Rightarrow y=-x-1$$$$\sqrt{1-x^2}-\sqrt{1-y^2}=0\Rightarrow 1-x^2=1-y^2$$$$1-x^2=1-{(-x-1)}^2$$$$1-x^2=-2x-x^2$$$$x=-\frac{1}{2}\Rightarrow y=-\frac{1}{2}\Rightarrow$$$$\Rightarrow \cos b=\cos c=-\frac{1}{2}\Rightarrow$$$$\Rightarrow b=c=\pm \frac{2\pi }{3}+2\pi z\wedge z\in \mathbb{Z}\mathrm{but}\sin b=-\sin c\Rightarrow$$$$a=\alpha \in \mathbb{R}$$$$b=\alpha \pm \frac{2\pi }{3}+2\pi z_1\wedge c=\alpha \mp \frac{2\pi }{3}+2\pi z_2\wedge z_1,z_2\in \mathbb{Z}$$

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