\( \dfrac{1}{\cos\theta} = 3\cos\theta + 1 \)
\( 1 = 3\cos^2\theta + \cos\theta \)
\( 3\cos^2\theta + \cos\theta - 1 = 0 \)

\( \cos\theta = \dfrac{-1 \pm \sqrt{1^2 - 4(3)(-1)}}{2 \times 3} \)
\( = \dfrac{-1 \pm \sqrt{13}}{6} \)

\( \cos\theta \approx 0.4342 \) or \( \cos\theta \approx -0.7675 \)

For \( \cos\theta = 0.4342 \): \( \theta = 64.3^\circ \) and \( 360^\circ - 64.3^\circ = 295.7^\circ \)
For \( \cos\theta = -0.7675 \): \( \theta = 140.1^\circ \) and \( 360^\circ - 140.1^\circ = 219.9^\circ \)

\(\boxed{\theta = 64.3^\circ,\; 140.1^\circ,\; 219.9^\circ,\; 295.7^\circ}\)