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Modulus and principal argument of a given complex number

This type of question, asked 3 times: 30 Jan 2023, 31 Jan 2023, 29 Jan 2024. All 3 questions below, oldest first.

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  1. Question 1

    30 Jan 2023

    81
    Let z=1+iz = 1 + i and z1=1+iz‾z‾(1−z)+1zz_{1}=\frac{1 + i\overline{z}}{\overline{z}(1 - z) +\frac{1}{z}}. Then 12πarg(z1)\frac{12}{\pi}arg\left( z_{1} \right) is equal to
  2. Question 2

    31 Jan 2023

    154
    The complex number z=i−1cos⁡π3+isinπ3z =\frac{i - 1}{\cos\frac{\pi}{3}+ isin\frac{\pi}{3}} is equal to :

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  3. Question 3

    29 Jan 2024

    156
    Let rr and θ\theta respectively be the modulus and amplitude of the complex number z=2−i(2tan⁡5π8)z = 2 - i\left( 2\tan\frac{5\pi}{8} \right), then (r,θ)(r,\theta) is equal to

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