First, multiply both sides by \(2 + \cos \theta\) to eliminate the fraction:

\(13 \sin^2 \theta + \cos \theta (2 + \cos \theta) = 2(2 + \cos \theta)\)

Simplify the equation:

\(13 \sin^2 \theta + 2 \cos \theta + \cos^2 \theta = 4 + 2 \cos \theta\)

Rearrange terms:

\(13 \sin^2 \theta + \cos^2 \theta = 4\)

Use the identity \(\sin^2 \theta = 1 - \cos^2 \theta\):

\(13(1 - \cos^2 \theta) + \cos^2 \theta = 4\)

Simplify:

\(13 - 13 \cos^2 \theta + \cos^2 \theta = 4\)

\(13 - 12 \cos^2 \theta = 4\)

\(12 \cos^2 \theta = 9\)

\(\cos^2 \theta = \frac{3}{4}\)

\(\cos \theta = \pm \frac{\sqrt{3}}{2}\)

Thus, \(\theta = 30^\circ, 150^\circ\).