Start with the equation \(8 \sin^2 \theta + 6 \cos \theta + 1 = 0\).

Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), substitute to get:

\(8(1 - \cos^2 \theta) + 6 \cos \theta + 1 = 0\).

Simplify to:

\(8 - 8 \cos^2 \theta + 6 \cos \theta + 1 = 0\).

Combine like terms:

\(-8 \cos^2 \theta + 6 \cos \theta + 9 = 0\).

Rearrange to:

\(8 \cos^2 \theta - 6 \cos \theta - 9 = 0\).

Factor the quadratic equation:

\((4 \cos \theta + 3)(2 \cos \theta - 3) = 0\).

Set each factor to zero:

\(4 \cos \theta + 3 = 0\) or \(2 \cos \theta - 3 = 0\).

Solve for \(\cos \theta\):

\(\cos \theta = -\frac{3}{4}\).

Find \(\theta\) using the inverse cosine function:

\(\theta = \cos^{-1}(-0.75)\).

Within the range \(0^\circ < \theta < 180^\circ\), \(\theta = 138.6^\circ\).