$\left| {{\mkern 1mu} \begin{array}{*{20}{c}} {1 + {{\cos }^2}\theta }&{{{\sin }^2}\theta }&{4\sin 3\theta }\\ {{{\cos }^2}\theta }&{1 + {{\sin }^2}\theta }&{4\sin 3\theta }\\ {{{\cos }^2}\theta }&{{{\sin }^2}\theta }&{1 + 4\sin 3\theta } \end{array}{\mkern 1mu} } \right| = 0$

$\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}$

$\left| {{\mkern 1mu} \begin{array}{*{20}{c}} 2&{{{\sin }^2}\theta }&{4\sin 3\theta }\\ 2&{1 + {{\sin }^2}\theta }&{4\sin 3\theta }\\ 1&{{{\sin }^2}\theta }&{1 + 4\sin 3\theta } \end{array}{\mkern 1mu} } \right| = 0$

$\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}, \mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-\mathrm{R}_{3}$

$\left| {{\mkern 1mu} \begin{array}{*{20}{c}} 0&{ - 1}&0\\ 1&1&{ - 1}\\ 1&{{{\sin }^2}\theta }&{1 + 4\sin 3\theta } \end{array}{\mkern 1mu} } \right| = 0$

or $4 \sin 3 \theta=-2$

$\sin 3 \theta=-\frac{1}{2}$

$\theta=\frac{7 \pi}{18}$