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Home » Divide the entire equation by 6: - wispro

Divide the entire equation by 6: - wispro

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 z = e^{-2\pi i/9} \cdot e^{2\pi i k / 3} = e^{-2\pi i (1 - 3k)/9} = e^{2\pi i (2/9 + k/3)} = e^{2\pi i (2 + 3k)/9}, \quad k = 0,1,2

📌 So all roots have arguments:

📌 From first set: \( \theta = \frac{2\pi(3k+1)}{9} \), \( k=0,1,2 \) → \( \frac{2\pi}{9}, \frac{8\pi}{9}, \frac{14\pi}{9} \)

📌 From second set: \( \theta = \frac{2\pi(3k+2)}{9} \), \( k=0,1,2 \) → \( \frac{4\pi}{9}, \frac{10\pi}{9}, \frac{16\pi}{9} \)

📌 List the real parts: \( \cos\theta \) for each:

📌 \( \cos\left(\frac{2\pi}{9}\right) \approx \cos(40^\circ) \approx 0.766 \)

📌 \( \cos\left(\frac{4\pi}{9}\right) = \cos(80^\circ) \approx 0.1736 \)

📌 \( \cos\left(\frac{8\pi}{9}\right) = \cos(160^\circ) \approx -0.9397 \)

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