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Home » \( 220 = n(3n + 7) \) - Network4

\( 220 = n(3n + 7) \) - Network4

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 \frac{1}{\frac{1}{2} \sin(2x)} = 2\sqrt{2} \Rightarrow \frac{2}{\sin(2x)} = 2\sqrt{2} \Rightarrow \frac{1}{\sin(2x)} = \sqrt{2} \Rightarrow \sin(2x) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}.

📌 Now solve $ \sin(2x) = \frac{\sqrt{2}}{2} $ for $ x \in (0, \pi) $. Then $ 2x \in (0, 2\pi) $.

📌 The solutions to $ \sin \theta = \frac{\sqrt{2}}{2} $ in $ (0, 2\pi) $ are:

📌 \theta = \frac{\pi}{4},\ \frac{3\pi}{4}.

📌 So $ 2x = \frac{\pi}{4} \Rightarrow x = \frac{\pi}{8} $, and $ 2x = \frac{3\pi}{4} \Rightarrow x = \frac{3\pi}{8} $.

📌 Both values lie in $ (0, \pi) $.

📌 Therefore, the solutions are $ \boxed{\frac{\pi}{8},\ \frac{3\pi}{8}} $.

📌 Question: A social media strategist models engagement growth on a scientific platform using the complex function $ z^6 + z^3 + 1 = 0 $. Among all roots, find the maximum imaginary part, expressed as $ \sin \theta $ for $ \theta \in (0, \pi) $. Find $ \theta $.

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