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

For non-trivial solutions

$\left|\begin{array}{ccc}1& 3& 7\\ -1& 4& 7\\ \mathrm{s}\mathrm{i}\mathrm{n}3\theta & \mathrm{c}\mathrm{o}\mathrm{s}2\theta & 2\end{array}\right|=0$

$\Rightarrow 4{\mathrm{s}\mathrm{i}\mathrm{n}}^3\theta +4{\mathrm{s}\mathrm{i}\mathrm{n}}^2\theta -3\mathrm{s}\mathrm{i}\mathrm{n}\theta =0$

$\Rightarrow \mathrm{s}\mathrm{i}\mathrm{n}\theta =0$ or $\mathrm{s}\mathrm{i}\mathrm{n}\theta =\frac{1}{2}$ or $\mathrm{s}\mathrm{i}\mathrm{n}\theta =-\frac{3}{2}$

Since $\theta \in \left(0,\pi \right)$

$\Rightarrow \theta =\frac{\pi }{6}$ and $\frac{5\pi }{6}$ will satisfy the equation

because $\sin \theta =-\frac{3}{2}$ is not possible as range of $\sin \theta$ is $\left[-1,1\right]$

$\Rightarrow$ Number of $\theta$ ' $s=2$