Matrices & Determinants · Homogeneous system non-trivial solution

The sum of all possible values of

\theta \in [0, 2\pi]

, for which the system of equations:

\begin{align*} x\cos3\theta - 8y - 12z &= 0\\ x\cos2\theta + 3y + 3z &= 0\\ x + y + 3z &= 0 \end{align*}

has a non-trivial solution, is equal to:

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