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

["Title: All Angles ( z \in [0^\circ, 360^\circ] ) that Satisfy ( 2\sin(2z) = \sqrt{3} )", "---", "## Introduction", "Solving trigonometric equations like ( 2\sin(2z) = \sqrt{3} ) can seem challenging at first, but with the right approach, finding all solutions in the interval ([0^\circ, 360^\circ]) becomes straightforward. This article explains step-by-step how to find all angles ( z ) that satisfy the equation ( 2\sin(2z) = \sqrt{3} ), ensuring clarity and accuracy for students and math enthusiasts alike.", "---", "## Step 1: Simplify the Equation", "Start by isolating the sine function:\n[\n2\sin(2z) = \sqrt{3} \implies \sin(2z) = \frac{\sqrt{3}}{2}\n]", "---", "## Step 2: Solve for the Angle ( 2z )", "We know from trigonometric values that:\n[\n\sin(\ heta) = \frac{\sqrt{3}}{2} \quad \ ext{when} \quad \ heta = 60^\circ \quad \ ext{or} \quad \ heta = 120^\circ\n]\nwithin one full rotation ([0^\circ, 360^\circ]).", "Since our argument is ( 2z ), set up:\n[\n2z = 60^\circ \quad \ ext{or} \quad 2z = 120^\circ\n]", "But sine is periodic with period ( 360^\circ ), so general solutions include:\n[\n2z = 60^\circ + 360^\circ k \quad \ ext{or} \quad 2z = 120^\circ + 360^\circ k \quad (k \in \mathbb{Z})\n]", "---", "## Step 3: Solve for ( z ) in ([0^\circ, 360^\circ])", "We seek values of ( z ) in ([0^\circ, 360^\circ]), so consider ( k = 0 ) and ( k = 1 ) to capture all solutions within range.", "### Case 1: ( 2z = 60^\circ + 360^\circ k )\nFor ( k = 0 ):\n[\n2z = 60^\circ \implies z = 30^\circ\n]\nFor ( k = 1 ):\n[\n2z = 420^\circ \implies z = 210^\circ\n]\n(larger than 360°, discard)", "### Case 2: ( 2z = 120^\circ + 360^\circ k )\nFor ( k = 0 ):\n[\n2z = 120^\circ \implies z = 60^\circ\n]\nFor ( k = 1 ):\n[\n2z = 480^\circ \implies z = 240^\circ\n]\n(larger than 360°, discard)", "---", "## Step 4: Verify All Solutions", "Check that all candidate angles lie in ([0^\circ, 360^\circ]) and satisfy the original equation:", "- For ( z = 30^\circ ):\n ( 2z = 60^\circ \implies \sin(60^\circ) = \frac{\sqrt{3}}{2} \Rightarrow 2\sin(60^\circ) = \sqrt{3} ) ✓", "- For ( z = 60^\circ ):\n ( 2z = 120^\circ \implies \sin(120^\circ) = \frac{\sqrt{3}}{2} \Rightarrow 2\sin(120^\circ) = \sqrt{3} ) ✓", "- For ( z = 210^\circ ):\n ( 2z = 420^\circ \equiv 60^\circ \ (\ ext{mod } 360^\circ) \Rightarrow \sin(420^\circ) = \sin(60^\circ) \Rightarrow 2\sin(420^\circ) = \sqrt{3} ) ✓", "- For ( z = 240^\circ ):\n ( 2z = 480^\circ \equiv 120^\circ \ (\ ext{mod } 360^\circ) \Rightarrow \sin(480^\circ) = \sin(120^\circ) \Rightarrow 2\sin(480^\circ) = \sqrt{3} ) ✓", "All four values satisfy the equation.", "---", "## Final Answer", "The complete set of solutions for angles ( z \in [0^\circ, 360^\circ] ) that satisfy ( 2\sin(2z) = \sqrt{3} ) is:\n[\n\boxed{30^\circ,\ 60^\circ,\ 210^\circ,\ 240^\circ}\n]", "---", "## Bonus: Graphical Interpretation", "Plotting ( 2\sin(2z) ) over ([0^\circ, 360^\circ]) reveals two sinusoidal peaks, each reaching ( \sqrt{3} ) at ( z = 30^\circ, 60^\circ, 210^\circ, 240^\circ ), confirming visual accuracy.", "---", "## Why This Matters", "Understanding how to solve such equations strengthens foundational trigonometry skills used in physics, engineering, and signal processing. Mastering angle transformations and periodic behavior is essential for tackling advanced problems.", "---", "Keywords: solve (2\sin(2z) = \sqrt{3}), find (z \in [0^\circ, 360^\circ]), trigonometric equation solutions, sine function angles, angle measurement, periodic functions, math tutorial.", "---", "Hope this guide helps you confidently solve similar trigonometric equations!"]