Question

If $a=e^{i\alpha },b=e^{i\beta },c=e^{i\gamma }$ and $\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma$, then prove the following
a+b+c = 0

Answer 1

$cos\alpha +cos\beta +cos\gamma =0................\left(1\right)sin\alpha +sin\beta +sin\gamma =0................\left(2\right)multiplying\left(2\right)\times \left(1\right)andaddingto\left(1\right)(cos\alpha +isin\alpha )+(cos\beta +isin\beta )+(cos\gamma +isin\gamma )=0accordingtoDeMour{e}^{\prime }stheorem,cos\alpha +isin\alpha =e^{i\alpha }=acos\beta +isin\beta =e^{i\beta }=bcos\gamma +isin\gamma =e^{i\gamma }=c\therefore (a+b+c)=0$