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Home » Let integers: \(x\) and \(x + 2\) - Portal da AcÃºstica

Let integers: \(x\) and \(x + 2\) - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 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.

📌 \sin(2z) + \sqrt{3} \cos(2z) = 1

📌 We rewrite the left-hand side as a single sine function. Let:

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