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Home » To find the radius \( r \), use the formula for circumference: - Aurero

To find the radius \( r \), use the formula for circumference: - Aurero

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 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} $.

📌 We now solve $ z^3 = \omega $ and $ z^3 = \omega^2 $. Each has three cube roots.

📌 First, write $ \omega = e^{2\pi i / 3} $, so the cube roots are:

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