Solution: The equation $ \sin(2\theta) = \frac{\sqrt{3}}{2} $ implies $ 2\theta = 60^\circ + 360^\circ k $ or $ 2\theta = 120^\circ + 360^\circ k $ for integer $ k $. Solving for $ \theta $, we get $ \theta = 30^\circ + 180^\circ k $ or $ \theta = 60^\circ + 180^\circ k $. Within $ [0^\circ, 360^\circ] $, the solutions are $ 30^\circ, 60^\circ, 210^\circ, 240^\circ $. Thus, the answer is $ \boxed{30^\circ}, \boxed{60^\circ}, \boxed{210^\circ}, \boxed{240^\circ} $.

["Solving $ \sin(2\ heta) = \frac{\sqrt{3}}{2} $: Step-by-Step Explanation & Solutions in $ [0^\circ, 360^\circ] $", "Trigonometric equations like $ \sin(2\ heta) = \frac{\sqrt{3}}{2} $ appear frequently in mathematics, physics, and engineering. Understanding how to solve such equations efficiently not only improves problem-solving skills but also deepens comprehension of periodic functions. In this article, we break down the solution to $ \sin(2\ heta) = \frac{\sqrt{3}}{2} $ step by step, exploring its implications and verifying all valid solutions within the interval $ [0^\circ, 360^\circ] $.", "### Understanding the Equation", "The sine function equals $ \frac{\sqrt{3}}{2} $ at specific standard angles. Recall the general solution for $ \sin x = \frac{\sqrt{3}}{2} $:", "$$\nx = 60^\circ + 360^\circ k \quad \ ext{or} \quad x = 120^\circ + 360^\circ k\n$$", "Since the equation involves $ 2\ heta $ instead of $ \ heta $, let:", "$$\nx = 2\ heta\n$$", "Then the equation becomes:", "$$\n\sin x = \frac{\sqrt{3}}{2} \quad \Rightarrow \quad 2\ heta = 60^\circ + 360^\circ k \quad \ ext{or} \quad 2\ heta = 120^\circ + 360^\circ k\n$$", "### Solving for $ \ heta $", "To isolate $ \ heta $, divide both sides by 2:", "$$\n\ heta = \frac{60^\circ + 360^\circ k}{2} = 30^\circ + 180^\circ k\n$$\n$$\n\ heta = \frac{120^\circ + 360^\circ k}{2} = 60^\circ + 180^\circ k\n$$", "This gives the general solution:", "$$\n\ heta = 30^\circ + 180^\circ k \quad \ ext{or} \quad \ heta = 60^\circ + 180^\circ k\n$$", "### Restricting to $ [0^\circ, 360^\circ] $", "We now find all values of $ \ heta $ within $ [0^\circ, 360^\circ] $ by testing integer values of $ k $.", "- For $ \ heta = 30^\circ + 180^\circ k $:\n - $ k = 0 $ → $ \ heta = 30^\circ $\n - $ k = 1 $ → $ \ heta = 210^\circ $\n - $ k = 2 $ → $ \ heta = 390^\circ $ (exceeds 360°, discard)", "- For $ \ heta = 60^\circ + 180^\circ k $:\n - $ k = 0 $ → $ \ heta = 60^\circ $\n - $ k = 1 $ → $ \ heta = 240^\circ $\n - $ k = 2 $ → $ \ heta = 420^\circ $ (exceeds 360°, discard)", "### Final Solutions", "Therefore, the complete set of solutions in the desired interval is:", "$$\n\boxed{30^\circ}, \quad \boxed{60^\circ}, \quad \boxed{210^\circ}, \quad \boxed{240^\circ}\n$$", "### Why This Method Works", "By reducing the original equation using substitution and known sine values, we simplify it into a standard form. Recognizing that $ \sin x = \frac{\sqrt{3}}{2} $ occurs at $ 60^\circ $ and $ 120^\circ $ allows us to build a complete solution set quickly. Dividing by 2 accounts for the doubled angle, and careful bounds ensure all solutions lie within the target angular range.", "### Conclusion", "Equation-solving in trigonometry benefits greatly from recognizing reference angles and periodicity. Whether in academics, test preparation, or applied sciences, mastering such techniques leads to clearer insights and faster problem resolution. The equation $ \sin(2\ heta) = \frac{\sqrt{3}}{2} $ serves as an excellent example of applying foundational trigonometric principles to find precise, accurate answers.", "---", "Keywords: $ \sin(2\ heta) = \frac{\sqrt{3}}{2} $, solving trigonometric equations, $ 2\ heta $ solutions, angle finding, $ [0^\circ, 360^\circ] $, $ \ heta = 30^\circ, 60^\circ, 210^\circ, 240^\circ $"]