• # question_answer If $\cos \alpha +\cos \beta +\cos \gamma =\sin \alpha +\sin \beta +\sin \gamma =0$ then $\cos 3\alpha +\cos 3\beta +\cos 3\gamma$ equals to  [Karnataka CET 2000] A) 0 B) $\cos (\alpha +\beta +\gamma )$ C) $3\cos (\alpha +\beta +\gamma )$ D) $3\sin (\alpha +\beta +\gamma )$

$\cos \alpha +\cos \beta +\cos \gamma =0$ and $\sin \alpha +\sin \beta +\sin \gamma =0$ Let $a=\cos \alpha +i\sin \alpha \,;\,b=\cos \beta +i\sin \beta$ and $c=\cos \gamma +i\sin \gamma .$ Therefore $a+b+c=(\cos \alpha +\cos \beta +\cos \gamma )$$+i\,(\sin \alpha +\sin \beta +\sin \gamma )$            $=0+i0=0$ If $a+b+c=0,$ then ${{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc$ or ${{(\cos \alpha +i\,\sin a)}^{3}}+{{(\cos \beta +i\sin \beta )}^{3}}+{{(\cos \gamma +i\sin \gamma )}^{3}}$$=3(\cos \alpha +i\sin \alpha )\,(\cos \beta +i\sin \beta )\,(\cos \gamma +i\sin \gamma )$ $\Rightarrow (\cos 3\alpha +i\sin 3\alpha )+(\cos 3\beta +i\sin 3\beta )+(\cos 3\gamma +i\sin 3\gamma )$$=3[\cos (\alpha +\beta +\gamma )+i\sin (\alpha +\beta +\gamma )]$ or $\cos 3\alpha +\cos 3\beta +\cos 3\gamma =3\cos (\alpha +\beta +\gamma ).$