For a homogeneous system of equations to have a non-trivial solution, the determinant of the coefficient matrix must be zero.\n\n\[ \begin{vmatrix} 1 & \sin\theta & \cos\theta \\ 1 & \cos\theta & \sin\theta \\ 1 & -\sin\theta & -\cos\theta \end{vmatrix} = 0 \]\n\nExpanding the determinant along the first column:\n\[ 1(-\cos^2\theta + \sin^2\theta) - 1(-\sin\theta\cos\theta + \sin\theta\cos\theta) + 1(\sin^2\theta - \cos^2\theta) = 0 \]\n\[ -\cos 2\theta - 0 - \cos 2\theta = 0 \]\n\[ -2\cos 2\theta = 0 \]\n\[ \cos 2\theta = 0 \]\n\nSince \( \theta \in [0, 2\pi] \), \( 2\theta \in [0, 4\pi] \).\nThe solutions for \( 2\theta \) are \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2} \).\n\nThus, \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4} \).\n\nThere are 4 values of \( \theta \).