Solution: Divide both sides by 2: $\sin(2z) = \frac{\sqrt{3}}{2}$. The general solutions for $2z$ are $2z = 60^\circ + 360^\circ k$ or $2z = 120^\circ + 360^\circ k$, where $k$ is an integer. Solve for $z$: $z = 30^\circ + 180^\circ k$ or $z = 60^\circ + 180^\circ k$. Within $[0^\circ, 360^\circ]$, valid solutions are $z = 30^\circ, 60^\circ, 210^\circ, 240^\circ$. Final answer: $\boxed{30^\circ, 60^\circ, 210^\circ, 240^\circ}$.

["solving $\sin(2z) = \frac{\sqrt{3}}{2}$: General and Specific Solutions", "Trigonometric equations like $\sin(2z) = \frac{\sqrt{3}}{2}$ often arise in advanced algebra and precalculus. Mastering their solutions helps solve periodic problems accurately. This article details the step-by-step solution to $\sin(2z) = \frac{\sqrt{3}}{2}$, delivering clear, practical insights for students and math enthusiasts.", "---", "### Understanding the Equation", "We begin with:\n$$\n\sin(2z) = \frac{\sqrt{3}}{2}\n$$\nThis equation identifies the sine of angle $2z$ equal to $\frac{\sqrt{3}}{2}$, a well-known value with reference angles at $60^\circ$ and $120^\circ$.", "---", "### Step 1: Solve for $2z$", "Because sine is periodic, we find all general angles where $\sin(\ heta) = \frac{\sqrt{3}}{2}$:", "$$\n2z = 60^\circ + 360^\circ k \quad \ ext{or} \quad 2z = 120^\circ + 360^\circ k \quad \ ext{for any integer } k\n$$", "These represent all solutions repeating every full rotation ($360^\circ$).", "---", "### Step 2: Solve for $z$", "Divide both sides of each equation by 2 to isolate $z$:\n$$\nz = \frac{60^\circ + 360^\circ k}{2} = 30^\circ + 180^\circ k\n$$\n$$\nz = \frac{120^\circ + 360^\circ k}{2} = 60^\circ + 180^\circ k\n$$", "So, the general solution is:\n$$\nz = 30^\circ + 180^\circ k \quad \ ext{or} \quad z = 60^\circ + 180^\circ k\n$$", "---", "### Step 3: Restrict to $z \in [0^\circ, 360^\circ]$", "We now find all values of $k$ such that $z$ stays within the interval $[0^\circ, 360^\circ]$.", "For $z = 30^\circ + 180^\circ k$:\n- $k = 0 \Rightarrow z = 30^\circ$\n- $k = 1 \Rightarrow z = 210^\circ$\n- $k = 2 \Rightarrow z = 390^\circ > 360^\circ$ → excluded", "For $z = 60^\circ + 180^\circ k$:\n- $k = 0 \Rightarrow z = 60^\circ$\n- $k = 1 \Rightarrow z = 240^\circ$\n- $k = 2 \Rightarrow z = 420^\circ > 360^\circ$ → excluded", "---", "### Final Solutions in the Interval", "Combining valid values, the solutions in $[0^\circ, 360^\circ]$ are:\n$$\nz = 30^\circ, 60^\circ, 210^\circ, 240^\circ\n$$", "---", "### Summary", "$\sin(2z) = \frac{\sqrt{3}}{2}$ → general solution:\n$$\n2z = 60^\circ + 360^\circ k \quad \ ext{or} \quad 2z = 120^\circ + 360^\circ k\n\Rightarrow z = 30^\circ + 180^\circ k \quad \ ext{or} \quad z = 60^\circ + 180^\circ k\n$$\nWithin $[0^\circ, 360^\circ]$, the complete solution set is:\n$$\n\boxed{30^\circ, \ 60^\circ, \ 210^\circ, \ 240^\circ}\n$$", "Mastering these steps strengthens your ability to solve trigonometric equations efficiently and supports deeper understanding in calculus and physics applications."]