Question:** A biodiversity specialist is studying the periodic behavior of genetic expression modeled by the equation \(2\cos(2z) = \sqrt{3}\). Find the sum of all angles \(z \in [0^\circ, 360^\circ]\) that satisfy this equation.

["Title: Solving (2\cos(2z) = \sqrt{3}): Finding All Solutions and Their Sum in Degrees", "Meta Description: Discover how to solve the equation (2\cos(2z) = \sqrt{3}) across (z \in [0^\circ, 360^\circ]), including finding all solutions and their sum—valuable for biodiversity studies modeling genetic expression rhythms.", "---", "### Introduction", "Understanding periodic biological patterns is essential in biodiversity research, where gene expression often follows rhythmic cycles. A key mathematical tool in modeling these cycles involves trigonometric equations. One such equation is:", "[\n2\cos(2z) = \sqrt{3}\n]", "This article guides you through solving this equation for (z) within the interval ([0^\circ, 360^\circ]), identifies all valid solutions, and computes their sum—a process relevant in analyzing biological periodicity.", "---", "### Step 1: Simplify the Equation", "Start by isolating the cosine term:", "[\n\cos(2z) = \frac{\sqrt{3}}{2}\n]", "The cosine of an angle equals (\frac{\sqrt{3}}{2}) at standard angles. Recall:", "[\n\cos \ heta = \frac{\sqrt{3}}{2} \implies \ heta = 30^\circ \quad \ ext{and} \quad \ heta = 330^\circ \quad \ ext{(within } [0^\circ, 360^\circ)).\n]", "Since our variable is (2z), set:", "[\n2z = 30^\circ + 360^\circ k \quad \ ext{or} \quad 2z = 330^\circ + 360^\circ k, \quad \ ext{for integer } k.\n]", "---", "### Step 2: Solve for (z) in ([0^\circ, 360^\circ])", "We now solve for (z) by dividing the angles by 2:", "Case 1:\n[\n2z = 30^\circ \implies z = 15^\circ\n]\n[\n2z = 30^\circ + 360^\circ = 390^\circ \implies z = 195^\circ\n]\nBoth values are within ([0^\circ, 360^\circ]).", "Case 2:\n[\n2z = 330^\circ \implies z = 165^\circ\n]\n[\n2z = 330^\circ + 360^\circ = 690^\circ \implies z = 345^\circ\n]\nAgain, both (165^\circ) and (345^\circ) are valid.", "No further values of (k) yield (z) within the required interval.", "Thus, the three solutions are:", "[\nz = 15^\circ, 165^\circ, 195^\circ, 345^\circ\n]", "Note: All four values are valid within ([0^\circ, 360^\circ]), but verify:\n(2z) ranges from (0^\circ) to (720^\circ), and cosine period is (360^\circ), so full symmetry is captured.", "---", "### Step 3: Compute the Sum of Solutions", "Add all valid angles:", "[\n15^\circ + 165^\circ + 195^\circ + 345^\circ = (15 + 165) + (195 + 345) = 180 + 540 = 720^\circ\n]", "---", "### Significance in Biodiversity and Genetic Expression", "Modeling genetic oscillations using ( \cos(2z) ) helps identify cycles in gene expression linked to environmental or internal rhythms. Solving (2\cos(2z) = \sqrt{3}) reveals key time points (angles in degrees) where expression peaks align with the model's prediction. Summing these solutions aids quantitative analysis in ecological genomics, supporting deeper insights into biological periodicity.", "---", "### Conclusion", "The equation (2\cos(2z) = \sqrt{3}) has four solutions in ([0^\circ, 360^\circ]): (15^\circ), (165^\circ), (195^\circ), and (345^\circ). Their sum is:", "[\n\boxed{720^\circ}\n]", "This mathematical insight supports researchers studying rhythmic patterns in biodiversity at a molecular level.", "---", "Keywords: (2\cos(2z) = \sqrt{3}), solve trigonometric equation, periodic genetic expression, biodiversity, angle sum solution, cosine periodicity, mathematical modeling in biology, z in degrees.", "Tags: Gene expression modeling, trigonometric equations, biodiversity research, mathematical biology, cosine periodicity."]