Question: Compute the sum of all angles $x \in [0^\circ, 360^\circ]$ satisfying $\sin(2x) = \frac{\sqrt{3}}{2}$, relevant to the symmetry of a linguistic phoneme cycle.

["Computing the Sum of All Angles $x \in [0^\circ, 360^\circ]$ Where $\sin(2x) = \frac{\sqrt{3}}{2}$: A Symmetrical Exploration Relevant to Linguistic Phoneme Cycles", "---", "Understanding the Trigonometric Equation", "When solving trigonometric equations involving multiple angles, precision and pattern recognition are essential. One such problem concerns finding all angles $x$ in the interval $[0^\circ, 360^\circ]$ that satisfy the equation\n$$\n\sin(2x) = \frac{\sqrt{3}}{2}.\n$$\nThis equation is not only mathematically elegant but also symbolically resonant with natural symmetries—especially in fields like linguistics, where periodic cycles resemble phonemic structures.", "---", "Step 1: Solve the Inner Angle $2x$", "We begin by solving\n$$\n\sin(2x) = \frac{\sqrt{3}}{2}.\n$$\nThe sine function equals $\frac{\sqrt{3}}{2}$ at standard angles:\n$$\n2x = 60^\circ \quad \ ext{and} \quad 2x = 120^\circ\n$$\nwithin the base interval $[0^\circ, 360^\circ]$, since $\sin \ heta = \frac{\sqrt{3}}{2}$ at these two primary solutions per cycle.", "However, since $2x$ ranges from $0^\circ$ to $720^\circ$ as $x$ goes from $0^\circ$ to $360^\circ$, we must find all solutions in this expanded range.", "---", "Step 2: Find All Valid Solutions for $2x$", "The sine function has period $360^\circ$, so within $[0^\circ, 720^\circ]$, the solutions to $\sin \ heta = \frac{\sqrt{3}}{2}$ are:\n$$\n\ heta = 60^\circ, \quad 120^\circ, \quad 420^\circ, \quad 480^\circ.\n$$\nThese occur every $360^\circ$ plus the base solutions.", "---", "Step 3: Solve for $x$", "Divide each solution by 2 to recover $x$:\n- $2x = 60^\circ \Rightarrow x = 30^\circ$\n- $2x = 120^\circ \Rightarrow x = 60^\circ$\n- $2x = 420^\circ \Rightarrow x = 210^\circ$\n- $2x = 480^\circ \Rightarrow x = 240^\circ$", "Thus, the complete set of solutions in $[0^\circ, 360^\circ]$ is:\n$$\nx = 30^\circ,\ 60^\circ,\ 210^\circ,\ 240^\circ.\n$$", "---", "Step 4: Compute the Sum of All Solutions", "Add the angles:\n$$\n30^\circ + 60^\circ + 210^\circ + 240^\circ = 540^\circ.\n$$", "---", "Why This Symmetry Matters: A Linguistic Perspective", "In linguistics, phonemes often behave cyclically—like wave functions—forming periodic cycles of sound patterns. The equation $\sin(2x) = \frac{\sqrt{3}}{2}$ models a doubling of frequency, mimicking how linguistic rhythms repeat at doubled intensities or intervals (e.g., stress patterns, syllabic emphasis in speech).", "The solutions $x = 30^\circ, 60^\circ, 210^\circ, 240^\circ$ reflect a structured symmetry across the unit circle, analogous to balanced phonemic cycles where intervals repeat every $180^\circ$ (due to the $2x$ argument), illustrating a natural duality in phonological periodicity.", "This mathematical symmetry thus serves as a meaningful metaphor—and tool—for analyzing phoneme cycles, revealing deep connections between trigonometry and linguistic structure.", "---", "Conclusion", "The sum of all angles $x \in [0^\circ, 360^\circ]$ satisfying $\sin(2x) = \frac{\sqrt{3}}{2}$ is\n$$\n\boxed{540^\circ}.\n$$\nBeyond its mathematical value, this result illuminates the elegant symmetries underlying both trigonometric functions and linguistic cycles, demonstrating how periodic laws manifest across scientific and cultural domains."]