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

$\mathrm{a}=\cos \alpha +\mathrm{isin}\mathrm{\alpha },\mathrm{b}=\cos \mathrm{\beta }+\mathrm{isin}\mathrm{\beta },\mathrm{c}=\cos \mathrm{\gamma }+\mathrm{isin}\mathrm{\gamma }$
Then, $\mathrm{a}+2\mathrm{b}+3\mathrm{c}=(\cos \mathrm{\alpha }+2\cos \mathrm{\beta }+3\cos \mathrm{\gamma })+\mathrm{i}(\sin \mathrm{\alpha }+2\sin \mathrm{\beta }+3\sin \mathrm{\gamma })=0$
$\begin{array}{l}\Rightarrow {\mathrm{a}}^3+8{\mathrm{b}}^3+27{\mathrm{c}}^3=18\mathrm{abc}\\ \Rightarrow \cos 3\mathrm{\alpha }+8\cos 3\mathrm{\beta }+27\cos 3\mathrm{\gamma }=18\cos (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })\end{array}$
And $\sin 3\mathrm{\alpha }+8\sin 3\mathrm{\beta }+27\sin 3\mathrm{\gamma }=18\sin (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })$