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1980 VTRMC #8

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August 11, 2020
complex numbers

Problem Statement

Let z=x+iyz=x+iy be a complex number with xx and yy rational and with ∣z∣=1.|z| = 1.
(a) Find two such complex numbers. (b) Show that ∣z2n−1∣=2∣sin⁡nθ∣,|z^{2n}-1|=2|\sin n\theta|, where z=eiθ.z=e^{i\theta}. (c) Show that ∣z2n−1∣|z^2n -1| is rational for every n.n.
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