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Home » Substitute $ a = 15 $, $ b = 30 $ into (1): - Esdistancia

Substitute $ a = 15 $, $ b = 30 $ into (1): - Esdistancia

📅 March 17, 2026 👤 scraface

Mar 17, 2026

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📌 Using identity: \( 4\cos^3\theta - 3\cos\theta = \cos 3\theta \), but our equation is \( x^3 - 4x + 2 = 0 \).

📌 Try substitution: \( x = 2y \):

📌 Let \( y = \cos\theta \), then:

📌 4\cos^3\theta - 4\cos\theta + 1 = 0 \implies \cos 3\theta = \frac{1}{4}

📌 Using identity: \( \cos 3\theta = \frac{1}{4} \implies 3\theta = \pm \arccos\left(\frac{1}{4}\right) + 2k\pi \)

📌 \theta = \frac{1}{3} \arccos\left(\frac{1}{4}\right),\quad \frac{1}{3}\left(2\pi - \arccos\left(\frac{1}{4}\right)\right),\quad \frac{1}{3}\left(4\pi + \arccos\left(\frac{1}{4}\right)\right)

📌 So the three real solutions for \( y \) are:

📌 y_k = \frac{1}{3} \arccos\left(\frac{1}{4}\right) + \frac{2k\pi}{3},\quad k = 0,1,2

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