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Home » The Ultimate Christmas Gift Basket Ideas That Everyone’s Raving About This Year! - Upplift

The Ultimate Christmas Gift Basket Ideas That Everyone’s Raving About This Year! - Upplift

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 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 $.

📌 Thus, the expression models symmetric feedback in a periodic synthetic ecosystem and evaluates to:

📌 Question: Find all angles $ z \in [0^\circ, 360^\circ] $ such that $ \sin(2z) + \sqrt{3} \cos(2z) = 1 $, relevant to aligning phase shifts in a glaciological time-series model of basal sliding.

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