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Home » Calculate \( \sin(60^\circ) \approx 0.866 \). - Esdistancia

Calculate \( \sin(60^\circ) \approx 0.866 \). - Esdistancia

📅 March 17, 2026 👤 scraface

Mar 17, 2026

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📌 setzen wir die Vieta’schen Werte ein:

📌 \sum w_i^2 = 8^2 - 2 \cdot 9 = 64 - 18 = 46.

📌 Wir müssen überprüfen, ob alle Wurzeln \( w_i \) reell und positiv sind, da nur reelle positive \( w_i \) zu reellen \( v = w_i^2 \) führen. Wir analysieren das kubische Polynom \( f(w) = w^3 - 8w^2 + 9w - 18 \). Berechne Extremstellen:

📌 \( f'(w) = 3w^2 - 16w + 9 \). Diskriminante: \( 256 - 108 = 148 > 0 \), zwei kritische Punkte:

📌 \( w = \frac{16 \pm \sqrt{148}}{6} = \frac{16 \pm 2\sqrt{37}}{6} = \frac{8 \pm \sqrt{37}}{3} \).

📌 Approximiere \( \sqrt{37} \approx 6.08 \), also:

📌 \( w_1 \approx \frac{8 - 6.08}{3} \approx 0.64 \), \( w_2 \approx \frac{8 + 6.08}{3} \approx 4.69 \).

📌 Berechne \( f(0.64) \approx (0.64)^3 - 8(0.64)^2 + 9(0.64) - 18 \approx 0.26 - 3.26 + 5.76 - 18 \approx -15.24 < 0 \),

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