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Home » You Won’t Believe These Hidden ‘Hello’ Phrases in Other Languages! - Silent Sales Machine

You Won’t Believe These Hidden ‘Hello’ Phrases in Other Languages! - Silent Sales Machine

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 \boxed{\text{No solution}}

📌 Question: Among all the roots of \(z^8 + z^4 + 1 = 0\), the maximum imaginary part can be expressed as \(\sin \phi\), where \(0^\circ < \phi < 90^\circ\). Find \(\phi\) in degrees.

📌 Consider the equation:

📌 z^8 + z^4 + 1 = 0

📌 Let \(w = z^4\), then:

📌 w = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} = \frac{-1 \pm i\sqrt{3}}{2}

📌 So \(z^4 = \frac{-1 \pm i\sqrt{3}}{2} = e^{\pm 2\pi i/3}\), since these are the primitive cube roots of unity (excluding 1).

📌 Thus, the solutions for \(z\) are the 4th roots of \(e^{2\pi i/3}\)

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