#### 298.29Question: Find the sum of all angles $ z \in [0^\circ, 360^\circ] $ that satisfy $ 2\sin(2z) + 1 = 0 $.

["Finding the Sum of All Angles $ z \in [0^\circ, 360^\circ] $ That Satisfy $ 2\sin(2z) + 1 = 0 $", "If you’re searching for a concise and accurate solution to the equation $ 2\sin(2z) + 1 = 0 $ over the interval $ z \in [0^\circ, 360^\circ] $, you're in the right place. This article explains step-by-step how to solve the trigonometric equation, finds all valid angle solutions, and computes their sum — all while optimizing for relevance and SEO.", "---", "### Understanding the Equation", "We begin with the equation:\n$$\n2\sin(2z) + 1 = 0\n$$", "Subtracting 1 and dividing by 2, we simplify:\n$$\n\sin(2z) = -\frac{1}{2}\n$$", "This is the key trigonometric condition we must solve.", "---", "### Step 1: Solve for $ 2z $ in Degrees", "We seek all values of $ 2z $ such that:\n$$\n\sin(2z) = -\frac{1}{2}\n$$", "Recall that sine equals $-\frac{1}{2}$ at standard reference angles of $ 210^\circ $ and $ 330^\circ $ within one full turn ($ 0^\circ $ to $ 360^\circ $), and since sine has a period of $ 360^\circ $, we include all coterminal angles.", "So the general solutions for $ 2z $ are:\n$$\n2z = 210^\circ + 360^\circ k \quad \ ext{or} \quad 2z = 330^\circ + 360^\circ k \quad \ ext{for integer } k\n$$", "---", "### Step 2: Find Solutions for $ z \in [0^\circ, 360^\circ] $", "Since $ z \in [0^\circ, 360^\circ] $, then $ 2z \in [0^\circ, 720^\circ] $. We now find all values of $ 2z $ within $ [0^\circ, 720^\circ] $ that satisfy $ \sin(2z) = -\frac{1}{2} $.", "#### First cycle: $ k = 0 $\n- $ 2z = 210^\circ $ → $ z = 105^\circ $\n- $ 2z = 330^\circ $ → $ z = 165^\circ $", "Both values are in $ [0^\circ, 360^\circ] $.", "#### Second cycle: $ k = 1 $\n- $ 2z = 210^\circ + 360^\circ = 570^\circ $ → $ z = 285^\circ $\n- $ 2z = 330^\circ + 360^\circ = 690^\circ $ → $ z = 345^\circ $", "Both $ 285^\circ $ and $ 345^\circ $ are within $ [0^\circ, 360^\circ] $.", "#### $ k = -1 $ or higher values produce $ 2z < 0^\circ $ or $ > 720^\circ $, so invalid.", "No more valid solutions.", "---", "### Step 3: List All Valid Solutions", "The complete set of solutions is:\n$$\nz = 105^\circ, ; 165^\circ, ; 285^\circ, ; 345^\circ\n$$", "---", "### Step 4: Compute the Sum", "$$\n105^\circ + 165^\circ + 285^\circ + 345^\circ = ?\n$$", "Break it down:\n- $ 105 + 165 = 270 $\n- $ 285 + 345 = 630 $\n- $ 270 + 630 = 900 $", "Thus, the sum is $ 900^\circ $", "---", "### Final Answer", "The sum of all angles $ z \in [0^\circ, 360^\circ] $ satisfying $ 2\sin(2z) + 1 = 0 $ is\n$$\n\boxed{900^\circ}\n$$", "---", "### SEO Keywords & Tips", "Optimized for search engine visibility, this article includes high-impact keywords like:\n- find all solutions to $ 2\sin(2z) + 1 = 0 $\n- sum of angles satisfying $ \sin(2z) = -\frac{1}{2} $\n- z in degrees interval $ [0^\circ, 360^\circ] $\n- trigonometric equation solutions sum", "The clear structure, step-by-step explanation, and final boxed result improve readability and increase chances of ranking for related academic and educational searches.", "---", "If you're studying trigonometric equations or preparing for exams, solving $ \sin(2z) = -\frac{1}{2} $ over a full circle is a fundamental skill — and knowing their sum saves time during timed tests!"]