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Home » \( h^5/2 = 12^5/2 - \frac28825\pi \cdot \frac52 = 12^5/2 - \frac28810\pi \) - Appfinity Technologies

\( h^5/2 = 12^5/2 - \frac28825\pi \cdot \frac52 = 12^5/2 - \frac28810\pi \) - Appfinity Technologies

📅 March 1, 2026 👤 scraface

Mar 01, 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).

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