If system of equations \(x \cos \theta - 8y - 12z = 0\), \(x \cos 2\theta + y + 3z = 0\), \(x + y + 3z = 0\) has non-trivial solution, then find sum of values of \(\theta\) (where \(\theta \in [0, 2\pi]\)).
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Always look for row or column operations to create zeros in a matrix before expanding the determinant. It significantly reduces the complexity of the trigonometric equation.