The value of $\theta$for which the system $\left(\sin 3\theta \right)x-y+z=0,$$\left(\cos 2\theta \right)x+4y+3z=0,$$2x+7y+7z=0$ has a non-trivial solution is   (where $\frac{\mathrm{\pi }}{2}<\theta <\pi ,(\pi =3.14))$

The system has non-trivial solution, if

$\begin{array}{l}\left|\begin{array}{ccc}\sin 3\theta & -1& 1\\ \cos 2\theta & 4& 3\\ 2& 7& 7\end{array}\right|=0\\ \Rightarrow \sin 3\theta \left(7\right)+\cos 2\theta \left(14\right)+2(-7)=0\\ \Rightarrow \sin 3\theta =2(1-\cos 2\theta )\\ \Rightarrow \sin \theta \left(3-4{\sin }^2\theta \right)=4{\sin }^2\theta \\ \Rightarrow 4{\sin }^2\theta +4\sin \theta -3=0\\ \Rightarrow (2\sin \theta -1)(2\sin \theta +3)=0\\ \Rightarrow \sin \theta =\frac{1}{2};\sin \theta =\frac{-3}{2}\text{ (No solution). }\end{array}$

$\begin{array}{l}\Rightarrow \theta =\frac{5\pi }{6} \left(\because \theta \in (\pi /2,\pi )\right)\\ =2.617.\\ \end{array}$