$The number of values of \theta in(0, \pi ) for which the system of linear equations$$x+3 y+7 z=0$,$-x+4 y+7 z=0$,  $\left(\sin 3\theta \right)x+\left(\cos 2\theta \right)y+2z=0$$\text{ has a non-trivial solution, is }$

Since, the system of linear equations has, non-trivial
solution then determinant of coefficient matrix =0

$\text{ i.e. }\left|\begin{array}{ccc}\sin 3\theta & \cos 2\theta & 2\\ 1& 3& 7\\ -1& 4& 7\end{array}\right|=0$

$\sin 3\theta (21-28)-\cos 2\theta (7+7)+2(4+3)=0$

$\sin 3\theta +2\cos 2\theta -2=0$

$3\sin \theta -4{\sin }^3\theta +2-4{\sin }^2\theta -2=0$

$4{\sin }^3\theta +4{\sin }^2\theta -3\sin \theta =0$

$\sin \theta \left(4{\sin }^2\theta +4\sin \theta -3\right)=0$

$\sin \theta \left(4{\sin }^2\theta +6\sin \theta -2\sin \theta -3\right)=0$

$\sin \theta \left[2\sin \theta \right(2\sin \theta -1)+3(2\sin \theta -1\left)\right]=0$

$\sin \theta (2\sin \theta -1)(2\sin \theta +3)=0$

$\sin \theta =0,\sin \theta =\frac{1}{2} \left(\because \sin \theta \ne -\frac{3}{2}\right)$

$\theta =\frac{\pi }{6},\frac{5\pi }{6}$

 $Hence, for two values of \theta system of equations has non trivial solution$