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how long does a college basketball game take - ECD Germany

📅 May 7, 2026 👤 scraface

May 07, 2026

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📌 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 $.

📌 Solution: Consider the equation:

📌 z^6 + z^3 + 1 = 0.

📌 Let $ w = z^3 $. Then:

📌 So $ z^3 = \omega $ or $ z^3 = \omega^2 $, where $ \omega = e^{2\pi i / 3} = \frac{-1 + i\sqrt{3}}{2} $, a primitive cube root of unity. The other root is $ \omega^2 = e^{-2\pi i / 3} $.

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