Question: Find the sum of all angles $z \in [0^\circ, 360^\circ]$ that satisfy $\sin(2z) = \frac{\sqrt{3}}{2}$.

["Title: Solve $\sin(2z) = \frac{\sqrt{3}}{2}$: Find All Solutions and Their Sum in Degrees | Sum of Angles $[0^\circ, 360^\circ]$", "---", "Introduction:\nUnderstanding trigonometric equations is key to mastering geometry and wave phenomena. One commonly encountered equation is $\sin(2z) = \frac{\sqrt{3}}{2}$. In this article, we’ll solve this equation for all angles $z$ in the interval $[0^\circ, 360^\circ]$, compute each solution, and find the sum of all such $z$ values. Perfect for students, teachers, or anyone exploring trigonometry at any level.", "---", "### Step 1: Understand the Equation\nWe begin with:\n$$\n\sin(2z) = \frac{\sqrt{3}}{2}\n$$\nThis is a standard sine equation, but the argument is $2z$, not $z$. This means we solve for $2z$ first before back-solving to $z$.", "We know from trigonometric identities:\n$$\n\sin \ heta = \frac{\sqrt{3}}{2} \Rightarrow \ heta = 60^\circ \ ext{ and } 120^\circ \ ext{ within } [0^\circ, 360^\circ]\n$$\nBut since $z \in [0^\circ, 360^\circ]$, then $2z$ ranges from $0^\circ$ to $720^\circ$. So we seek all solutions to:\n$$\n2z = 60^\circ, 120^\circ, 420^\circ, 480^\circ\n$$\nThese correspond to the full set of angles in $[0^\circ, 720^\circ]$ where sine equals $\frac{\sqrt{3}}{2}$.", "---", "### Step 2: Solve for $z$\nDivide each solution by 2 to recover $z$:\n$$\nz = \frac{60^\circ}{2} = 30^\circ\n$$\n$$\nz = \frac{120^\circ}{2} = 60^\circ\n$$\n$$\nz = \frac{420^\circ}{2} = 210^\circ\n$$\n$$\nz = \frac{480^\circ}{2} = 240^\circ\n$$", "All four solutions lie within $[0^\circ, 360^\circ]$, so they are valid.", "---", "### Step 3: Verify Each Solution\nLet’s plug each value back to confirm:\n- $\sin(2 \cdot 30^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$ ✅\n- $\sin(2 \cdot 60^\circ) = \sin(120^\circ) = \frac{\sqrt{3}}{2}$ ✅\n- $\sin(2 \cdot 210^\circ) = \sin(420^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$ ✅\n- $\sin(2 \cdot 240^\circ) = \sin(480^\circ) = \sin(120^\circ) = \frac{\sqrt{3}}{2}$ ✅", "All verified.", "---", "### Step 4: Sum All Valid Solutions\nNow, compute the sum:\n$$\n30^\circ + 60^\circ + 210^\circ + 240^\circ = 540^\circ\n$$", "---", "### Step 5: Why This Matters\nFinding the sum of all angle solutions in a trigonometric interval is useful in fields such as engineering, physics, and signal processing. For example, in wave interference problems or rotational symmetry, knowing the total angular coverage helps analyze periodic behavior.", "---", "### Conclusion:\nThe values of $z \in [0^\circ, 360^\circ]$ satisfying $\sin(2z) = \frac{\sqrt{3}}{2}$ are:\n$$\n30^\circ,\ 60^\circ,\ 210^\circ,\ 240^\circ\n$$\nTheir sum is:\n$$\n\boxed{540^\circ}\n$$", "Understanding these angle solutions enriches your grasp of trigonometric functions and periodic equations—essential tools in mathematics and applied sciences.", "---", "Keywords:\n$\sin(2z) = \frac{\sqrt{3}}{2}$, angle sum $z \in [0^\circ, 360^\circ]$, trigonometric solutions, sine equation, sum of angles, periodic functions, geometry and trigonometry tutorial", "Meta Description:\nSolve $\sin(2z) = \frac{\sqrt{3}}{2}$ and find all $z \in [0^\circ, 360^\circ]$ that satisfy it. Step-by-step solution with verification and sum of angles (total = $\boxed{540^\circ}$). Perfect for math students and educators."]