إذن، $\gcd(105, 165) = \boxed{15}$.السؤال: أوجد مجموع جميع الزوايا $z \in [0^\circ, 360^\circ]$ التي تحقق $\sin(2z) = \frac{\sqrt{3}}{2}$.

["How to Solve: Find the Sum of All Angles ( z \in [0^\circ, 360^\circ] ) That Satisfy ( \sin(2z) = \frac{\sqrt{3}}{2} )", "Understanding the Equation", "We are tasked with finding all angles ( z ) in the interval ([0^\circ, 360^\circ]) such that\n[\n\sin(2z) = \frac{\sqrt{3}}{2}.\n]", "This equation involves a sine function with an argument ( 2z ), meaning the variable appears doubled. To solve it, we first find all solutions for ( 2z ), then back-substitute to find ( z ), and finally compute the sum of all valid angles.", "---", "Step 1: Solve for ( 2z )", "Recall that ( \sin \ heta = \frac{\sqrt{3}}{2} ) at two standard angles in each full rotation:\n[\n\ heta = 60^\circ \quad \ ext{and} \quad \ heta = 120^\circ\n]\nwithin ( [0^\circ, 360^\circ] ). Since sine is periodic with period ( 360^\circ ), the general solution is:\n[\n2z = 60^\circ + 360^\circ k \quad \ ext{or} \quad 2z = 120^\circ + 360^\circ k, \quad k \in \mathbb{Z}.\n]", "---", "Step 2: Find all ( z ) in ( [0^\circ, 360^\circ] )", "We now solve for ( z ) by dividing each solution by 2:", "1. From ( 2z = 60^\circ + 360^\circ k ):\n[\nz = 30^\circ + 180^\circ k\n]\nCheck values of ( k ) such that ( z \in [0^\circ, 360^\circ] ):\n- ( k = 0 \Rightarrow z = 30^\circ )\n- ( k = 1 \Rightarrow z = 210^\circ )\n- ( k = 2 \Rightarrow z = 390^\circ > 360^\circ ) → invalid\nSo valid values: ( z = 30^\circ, 210^\circ )", "2. From ( 2z = 120^\circ + 360^\circ k ):\n[\nz = 60^\circ + 180^\circ k\n]\nCheck values of ( k ):\n- ( k = 0 \Rightarrow z = 60^\circ )\n- ( k = 1 \Rightarrow z = 240^\circ )\n- ( k = 2 \Rightarrow z = 420^\circ > 360^\circ ) → invalid\nSo valid values: ( z = 60^\circ, 240^\circ )", "All solutions:\n[\nz = 30^\circ, 60^\circ, 210^\circ, 240^\circ\n]", "---", "Step 3: Compute the Sum", "Add the angles:\n[\n30^\circ + 60^\circ + 210^\circ + 240^\circ = 540^\circ\n]", "---", "Final Answer:\n[\n\boxed{540^\circ}\n]", "Additional Note for SEO:\nThis article explores solving trigonometric equations involving double angles, specifically ( \sin(2z) = \frac{\sqrt{3}}{2} ), by identifying all valid angles in a standard interval and computing their sum. Perfect for students learning trigonometric solution methods, periodic functions, and angle sum calculations in degrees. Use keywords like (\sin(2z) = \frac{\sqrt{3}}{2}), find all solutions in ([0^\circ, 360^\circ]), sum of angles in trigonometry."]