Given the equation \(\sin 2x = 2 \cos 2x\).

Divide both sides by \(\cos 2x\) (assuming \(\cos 2x \neq 0\)) to get:

\(\tan 2x = 2\).

Find the general solution for \(2x\):

\(2x = \tan^{-1}(2) + k \cdot 180^\circ\), where \(k\) is an integer.

Calculate \(\tan^{-1}(2) \approx 63.4^\circ\).

Thus, \(2x = 63.4^\circ\) or \(2x = 243.4^\circ\) (adding \(180^\circ\)).

Divide by 2 to find \(x\):

\(x = 31.7^\circ\) or \(x = 121.7^\circ\).

Both solutions are within the given range \(0^\circ \leq x \leq 180^\circ\).