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Home » لذلك، الطول هو \( 2w = 12 \). - Midis

لذلك، الطول هو \( 2w = 12 \). - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 The maximum value is $ \boxed{350^\circ} $. This corresponds to the peak resonance angle in the glacial oscillation model when physical constraints limit the domain to this interval.

📌 \boxed{350^\circ}

📌 Question: Compute $ \left( \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} \right)^{14} + \left( \cos \frac{2\pi}{7} - i \sin \frac{2\pi}{7} \right)^{14} $, a quantity arising in modeling periodic synthetic microbial interactions in subglacial biofilm networks.

📌 Let $ z = \cos \frac{2\pi}{7} + i \sin \frac{2\pi}{7} = e^{i \frac{2\pi}{7}} $. Then its conjugate is $ \overline{z} = \cos \frac{2\pi}{7} - i \sin \frac{2\pi}{7} = e^{-i \frac{2\pi}{7}} $.

📌 z^{14} + \overline{z}^{14} = \left(e^{i \frac{2\pi}{7}}\right)^{14} + \left(e^{-i \frac{2\pi}{7}}\right)^{14} = e^{i 4\pi} + e^{-i 4\pi}

📌 Since $ e^{i 4\pi} = \cos(4\pi) + i \sin(4\pi) = 1 $, and similarly $ e^{-i 4\pi} = 1 $, we get:

📌 z^{14} + \overline{z}^{14} = 1 + 1 = 2

📌 Alternatively, recognizing $ e^{i 4\pi} = e^{i 0} = 1 $, the sum is $ 2 \cos(0) = 2 $.

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