1. **State the problem:** Solve the equation $$\sin \theta + 2 \sin \theta \cos \theta = 0$$ for $$\theta$$ in the range $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Write the equation:** $$\sin \theta + 2 \sin \theta \cos \theta = 0$$. 3. **Factor the equation:** Factor out $$\sin \theta$$: $$\sin \theta (1 + 2 \cos \theta) = 0$$. 4. **Set each factor to zero:** - $$\sin \theta = 0$$ - $$1 + 2 \cos \theta = 0$$ 5. **Solve $$\sin \theta = 0$$:** $$\sin \theta = 0$$ at $$\theta = 0^\circ, 180^\circ, 360^\circ$$ within the given range. 6. **Solve $$1 + 2 \cos \theta = 0$$:** $$2 \cos \theta = -1$$ $$\cos \theta = -\frac{1}{2}$$ 7. **Find $$\theta$$ where $$\cos \theta = -\frac{1}{2}$$:** This occurs at $$\theta = 120^\circ$$ and $$\theta = 240^\circ$$ within the range $$0^\circ$$ to $$360^\circ$$. 8. **Combine all solutions:** $$\theta = 0^\circ, 120^\circ, 180^\circ, 240^\circ, 360^\circ$$. **Final answer:** $$\boxed{0^\circ, 120^\circ, 180^\circ, 240^\circ, 360^\circ}$$