Start with the equation \(\sin 2x + 3 \cos 2x = 0\).

Rearrange to find \(\tan 2x\):

\(\tan 2x = -3\).

Find the angle \(2x\) using the inverse tangent function:

\(2x = 180^\circ - 71.6^\circ\) or \(2x = 360^\circ - 71.6^\circ\).

This gives \(2x = 108.4^\circ\) or \(2x = 288.4^\circ\).

Divide by 2 to solve for \(x\):

\(x = 54.2^\circ\) or \(x = 144.2^\circ\).