1. The problem is to solve the trigonometric equation $$\sin(x) = \frac{1}{2}$$ for all solutions. 2. From the unit circle, we know that $$\sin(x) = \frac{1}{2}$$ at two principal angles in the interval $$[0, 2\pi)$$: $$x = \frac{\pi}{6}$$ (30°) and $$x = \frac{5\pi}{6}$$ (150°). 3. The sine function is periodic with period $$2\pi$$, so the solutions repeat every $$2\pi$$ radians. 4. Therefore, the general solutions can be written as: $$ x = \frac{\pi}{6} + 2\pi n \quad \text{and} \quad x = \frac{5\pi}{6} + 2\pi n \quad \text{for any integer } n. $$ 5. Looking at the sine graph, the equation $$\sin(x) = \frac{1}{2}$$ has infinitely many solutions because the sine wave crosses the line $$y=\frac{1}{2}$$ infinitely many times as $$x$$ increases or decreases. 6. In summary, the complete set of solutions to $$\sin(x) = \frac{1}{2}$$ is all angles coterminal with $$\frac{\pi}{6}$$ and $$\frac{5\pi}{6}$$, expressed as: $$ x = \frac{\pi}{6} + 2\pi n \quad \text{or} \quad x = \frac{5\pi}{6} + 2\pi n, \quad n \in \mathbb{Z}. $$