1. Start with the equation \(\sin \theta = 2 \cos 2\theta + 1\).

2. Use the double angle formula for cosine: \(\cos 2\theta = 1 - 2\sin^2 \theta\).

3. Substitute into the equation: \(\sin \theta = 2(1 - 2\sin^2 \theta) + 1\).

4. Simplify: \(\sin \theta = 2 - 4\sin^2 \theta + 1\).

5. Rearrange to form a quadratic equation: \(4\sin^2 \theta + \sin \theta - 3 = 0\).

6. Solve the quadratic equation using the quadratic formula: \(\sin \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4, b = 1, c = -3\).

7. Calculate the discriminant: \(b^2 - 4ac = 1 + 48 = 49\).

8. Solve for \(\sin \theta\):

\(\sin \theta = \frac{-1 \pm 7}{8}\)

9. This gives \(\sin \theta = \frac{3}{4}\) or \(\sin \theta = -1\).

10. For \(\sin \theta = \frac{3}{4}\), find \(\theta\):

\(\theta = \sin^{-1}\left(\frac{3}{4}\right) \approx 48.6^\circ\) and \(180^\circ - 48.6^\circ = 131.4^\circ\).

11. For \(\sin \theta = -1\), \(\theta = 270^\circ\).

12. The solutions in the interval \(0^\circ \leq \theta \leq 360^\circ\) are \(\theta = 48.6^\circ, 131.4^\circ, 270^\circ\).