1. **State the problem:** Evaluate $\sin(2\pi + 30)$. 2. **Recall the sine addition formula and periodicity:** The sine function has a period of $2\pi$, meaning $\sin(\theta + 2\pi) = \sin(\theta)$ for any angle $\theta$. 3. **Apply periodicity:** $$\sin(2\pi + 30) = \sin(30)$$ 4. **Evaluate $\sin(30)$:** $\sin(30) = \frac{1}{2}$. 5. **Final answer:** $$\sin(2\pi + 30) = \frac{1}{2}$$ This means the sine of $2\pi + 30$ is the same as the sine of $30$, which is $\frac{1}{2}$.