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Home » D: $\sin\left(\frac5\pi8\right)$ - Crankk.io

D: $\sin\left(\frac5\pi8\right)$ - Crankk.io

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 z^3 + 8z^2 + \frac{64}{3}z + \frac{512}{27} -8z^2 - \frac{128}{3}z - \frac{512}{9} + 9z + 24 - 18

📌 = z^3 + \left(8 - 8\right)z^2 + \left(\frac{64 - 128 + 27}{3}\right)z + \left(\frac{512}{27} - \frac{1536}{27} + \frac{648}{27} - \frac{486}{27}\right)

📌 = z^3 + \left(\frac{-407}{3}\right)z + \left(\frac{512 - 1536 + 648 - 486}{27}\right) = z^3 - \frac{407}{3}z - \frac{1422}{27} = z^3 - \frac{407}{3}z - \frac{474}{9}

📌 = z^3 - \frac{407}{3}z - \frac{158}{3}

📌 Diskriminante eines kubischen \( z^3 + pz + q \): \( \Delta = -4p^3 - 27q^2 \)

📌 Hier: \( p = -\frac{407}{3} \), \( q = -\frac{158}{3} \)

📌 \Delta = -4\left(-\frac{407}{3}\right)^3 - 27\left(-\frac{158}{3}\right)^2 = -4\left(-\frac{67432643}{27}\right) - 27\cdot\frac{24964}{9}

📌 Besser: positives Vorzeichen:

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