Start with the equation:

\(\frac{\tan \theta + 3 \sin \theta + 2}{\tan \theta - 3 \sin \theta + 1} = 2\)

Multiply both sides by the denominator:

\(\tan \theta + 3 \sin \theta + 2 = 2(\tan \theta - 3 \sin \theta + 1)\)

Expand and simplify:

\(\tan \theta + 3 \sin \theta + 2 = 2 \tan \theta - 6 \sin \theta + 2\)

Rearrange terms:

\(2 \tan \theta - \tan \theta - 6 \sin \theta - 3 \sin \theta + 2 - 2 = 0\)

\(\tan \theta - 9 \sin \theta = 0\)

Multiply by \(\cos \theta\):

\(\sin \theta - 9 \sin \theta \cos \theta = 0\)

Factorize:

\(\sin \theta (1 - 9 \cos \theta) = 0\)

Set each factor to zero:

\(\sin \theta = 0\) or \(1 - 9 \cos \theta = 0\)

For \(\sin \theta = 0\), \(\theta = 0^\circ\).

For \(1 - 9 \cos \theta = 0\), \(\cos \theta = \frac{1}{9}\).

Calculate \(\theta\) using \(\cos^{-1}\):

\(\theta \approx 83.6^\circ\).

Thus, \(\theta = 0^\circ\) or \(83.6^\circ\) (only answers in the given range).