Consider the system of equations in $x, y, z$ as

$x \sin 3θ-y+z=0$

$x \cos 2θ+ 4y + 3z = 0$

$2x + 7y+7z=0$

If this system has a non-trivial solution, then for any integer n, values of θ are given

$\left(n+\frac{(-1)^n}{6}\right)π$

The given system of equations has a non-trivial solution.

$∴\begin{vmatrix}\sin 3θ&-1&1\\\cos 2θ&4&3\\2&7&7\end{vmatrix}=0$

$⇒7 \sin 3θ+ 14 \cos 2θ-14=0$

$⇒3 \sin θ-4 \sin^3 θ+2-4 \sin^2θ-2=0$

$⇒\sin θ (4 \sin^2 θ + 4 \sin θ - 3) = 0$

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

$⇒\sin θ =0$ or, $\sin θ =\frac{1}{2}⇒ θ=nπ$ or, $θ=nπ+(-1)^n\frac{π}{6}$