If $\cos \alpha +2\cos \beta +3\cos \gamma =0=\sin \alpha +2\sin \beta +3\sin \gamma$  then the value of $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma$  is

$Let x=\cos \alpha +i \sin \alpha =cis\alpha ; y=\cos \beta +i \sin \beta =cis\beta ; z = \cos \gamma +i \sin \gamma =cis\gamma$   then      $x+2y+3z= \left(\cos \alpha +2\cos \beta +3\cos \gamma \right) +i\left(\sin \alpha +2\sin \beta +3\sin \gamma \right) =0+i0 =0$

 $therefore x^3+8y^3+27z^3=18xyz$  

                  ${{\left(cis \alpha \right)}^3}^{ }+8{\left(cis\beta \right)}^3+27{\left(cis\gamma \right)}^3=18 \left(cis\alpha \right)\left(cis \beta \right)\left(cis \gamma \right)$

                  $\Rightarrow cis 3\alpha + 8 cis3\beta + 27 cis 3\gamma = 18 cis \left(\alpha +\beta +\gamma \right)$            $\left(by Demovier\text{'}s theorem\right)$

                on comparing imaginary parts , $\sin 3\alpha +8\sin 3\beta +27\sin 3\gamma =18\sin (\alpha +\beta +\gamma )$