A Geometric Triple of Complex Numbers

★ 7/10complex numberstrigonometryGeometric ProgressionCasework

Problem

Let a∈Ra \in \mathbb{R} and θ∈[0,2π)\theta \in [0, 2\pi), and set

z1=cos⁡θ+isin⁡θ,z2=sin⁡θ+icos⁡θ,z3=a(1−i).z_1 = \cos\theta + i\sin\theta, \qquad z_2 = \sin\theta + i\cos\theta, \qquad z_3 = a(1 - i).

Find the number of pairs (a,θ)(a, \theta) for which z1,z2,z3z_1, z_2, z_3 form a geometric progression.

Answer

Solution

Difficulty7/10
Topicscomplex numbers, trigonometry, Geometric Progression, Casework

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