Let $a=\cos \alpha+i \sin \alpha$
$\quad b=\cos \beta+i \sin \beta$
and $\mathrm{c}=\cos \gamma+i \sin \gamma$
Then, $a+2 b+3 c=(\cos \alpha+2 \cos \beta+3 \cos \gamma)$
$\quad+i(\sin \alpha+2 \sin \beta+3 \sin \gamma)=0$
$\Rightarrow \quad a^{3}+8 b^{3}+27 c^{3}=18 a b c$
$\left(\because\right.$ if $\left.x+y+z=0 \Rightarrow x^{3}+y^{3}+z^{3}=3 x y z\right)$
$\Rightarrow \quad \cos 3 \alpha+8 \cos 3 \beta+27 \cos 3 \gamma$
and $\quad \sin 3 \alpha+8 \sin 3 \beta+27 \sin 3 \gamma$
$\quad=18 \sin (\alpha+\beta+\gamma)$
$=18 \sin \pi=0$