Question

If $cos\alpha +cos\beta +cos\gamma =0=sin\alpha +sin\beta +sin\gamma$ then $cos2\alpha +cos2\beta +cos2\gamma$ equals

• A. $2cos(\alpha +\beta +\gamma )$
• B. $cos2(\alpha +\beta +\gamma )$
• C. 0
• D. 1

Answer 1

The correct option is C 0

$\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma$
Let $a=\cos \alpha +i\sin \alpha ,b=\cos \beta +i\sin \beta$ and $c=\cos \gamma +i\sin \gamma$
$a+b+c=0$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$
$\Rightarrow ab+bc+ca=0$
$\Rightarrow a^2+b^2+c^2=0$
$\therefore \cos 2\alpha +\cos 2\beta +\cos 2\gamma =0$
Hence, option C.