Start with the equation:

\(\frac{1}{\cos \theta} + 3 \sin \theta \tan \theta + 4 = 0\)

Multiply through by \(\cos \theta\) to eliminate the fraction:

\(1 + 3 \sin^2 \theta + 4 \cos \theta = 0\)

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

\(1 + 3(1 - \cos^2 \theta) + 4 \cos \theta = 0\)

Simplify:

\(1 + 3 - 3 \cos^2 \theta + 4 \cos \theta = 0\)

\(3 \cos^2 \theta - 4 \cos \theta - 4 = 0\)

Now solve \(3 \cos^2 \theta - 4 \cos \theta - 4 = 0\):

Let \(x = \cos \theta\), then:

\(3x^2 - 4x - 4 = 0\)

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):

\(x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3}\)

\(x = \frac{4 \pm \sqrt{16 + 48}}{6}\)

\(x = \frac{4 \pm \sqrt{64}}{6}\)

\(x = \frac{4 \pm 8}{6}\)

\(x = 2 \text{ or } x = -\frac{2}{3}\)

Since \(\cos \theta\) cannot be 2, use \(\cos \theta = -\frac{2}{3}\):

\(\theta = \cos^{-1}(-\frac{2}{3})\)

\(\theta \approx 131.8^\circ \text{ or } 228.2^\circ\)