1. **State the problem:** Solve the equation $$\cos 2\theta + \cos \theta + 1 = 0$$ for all values of $$\theta$$ in radians. 2. **Use the double-angle formula:** Recall that $$\cos 2\theta = 2\cos^2 \theta - 1$$. 3. **Substitute into the equation:** $$2\cos^2 \theta - 1 + \cos \theta + 1 = 0$$ 4. **Simplify the equation:** $$2\cos^2 \theta + \cos \theta = 0$$ 5. **Factor the equation:** $$\cos \theta (2\cos \theta + 1) = 0$$ 6. **Set each factor equal to zero:** - $$\cos \theta = 0$$ - $$2\cos \theta + 1 = 0$$ 7. **Solve each equation:** - For $$\cos \theta = 0$$, $$\theta = \frac{\pi}{2} + k\pi$$, where $$k$$ is any integer. - For $$2\cos \theta + 1 = 0$$, solve for $$\cos \theta$$: $$2\cos \theta + 1 = 0 \Rightarrow 2\cos \theta = -1 \Rightarrow \cos \theta = -\frac{1}{2}$$ 8. **Find $$\theta$$ for $$\cos \theta = -\frac{1}{2}$$:** $$\theta = \pm \frac{2\pi}{3} + 2k\pi$$, where $$k$$ is any integer. **Final answer:** $$\theta = \frac{\pi}{2} + k\pi \quad \text{or} \quad \theta = \pm \frac{2\pi}{3} + 2k\pi, \quad k \in \mathbb{Z}$$