1. **State the problem:** Solve the equation $\cos \theta + 1 = 0$ for all values of $\theta$ where $\theta$ is in radians. 2. **Rewrite the equation:** $$\cos \theta + 1 = 0$$ Subtract 1 from both sides: $$\cos \theta = -1$$ 3. **Recall the cosine function properties:** The cosine function has a range of $[-1,1]$ and is periodic with period $2\pi$. 4. **Find the values of $\theta$ where $\cos \theta = -1$:** Cosine equals $-1$ at $\theta = \pi$ plus any integer multiple of the period $2\pi$. 5. **Write the general solution:** $$\theta = \pi + 2k\pi$$ where $k$ is any integer ($k \in \mathbb{Z}$). 6. **Explanation:** This means the cosine function reaches $-1$ at $\pi$, $3\pi$, $5\pi$, etc., and also at $-\pi$, $-3\pi$, etc., covering all such angles by adding multiples of $2\pi$. **Final answer:** $$\boxed{\theta = \pi + 2k\pi, \quad k \in \mathbb{Z}}$$