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📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A computational biotechnologist is modeling the phase alignment of two synthetic gene oscillators represented as complex numbers on the unit circle, given by $ z_1 = \cos heta_1 + i \sin heta_1 $ and $ z_2 = \cos heta_2 + i \sin heta_2 $. If the angle between them, measured as the smallest positive argument of $ z_1 \overline{z_2} $, is $ 60^\circ $, find the value of $ \cos( heta_1 - heta_2) $.

📌 We are given that the angle between $ z_1 $ and $ z_2 $ is $ 60^\circ $, which corresponds to the smallest positive argument of $ z_1 \overline{z_2} $. Since both are on the unit circle, $ |z_1| = |z_2| = 1 $, and

📌 z_1 \overline{z_2} = (\cos heta_1 + i \sin heta_1)(\cos heta_2 - i \sin heta_2) = \cos( heta_1 - heta_2) + i \sin( heta_1 - heta_2).

📌 The argument of $ z_1 \overline{z_2} $ is $ heta_1 - heta_2 $ (modulo $ 360^\circ $), and the smallest positive argument is given to be $ 60^\circ $. Since cosine determines the angle uniquely within $ [0^\circ, 180^\circ] $ for this context (as it's the smallest angle between them), we have

📌 heta_1 - heta_2 \equiv 60^\circ \pmod{180^\circ},

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