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(a) Sketch the following set, M, in the complex plane: M = { EC || ≤ 1 and |zi| ≤ 1}. : (b) Find all solutions z E C of the equation z³ = 8i. You may leave your answers in polar or exponential form. (c) By writing the numbers 1+ i√√3 and 1 - i√√3 in polar or exponential form, determine all natural numbers n for which (1 + i√√3)" = (1 − i√√3)". 1+z (d) Consider complex numbers z, w such that = W. 1-2 i. Show that if zia where a is a real number, then w = 1. ii. Conversely, show that if w❘: =1 then has the form z = ia with a ЄR.

(a) Sketch the following set, M, in the complex plane: M = { EC || ≤ 1 and |zi| ≤ 1}. : (b) Find all solutions z E C of the equation z³ = 8i. You may leave your answers in polar or exponential form. (c) By writing the numbers 1+ i√√3 and 1 - i√√3 in polar or exponential form, determine all natural numbers n for which (1 + i√√3)" = (1 − i√√3)". 1+z (d) Consider complex numbers z, w such that = W. 1-2 i. Show that if zia where a is a real number, then w = 1. ii. Conversely, show that if w❘: =1 then has the form z = ia with a ЄR.

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Complex Numbers
Section4.3: The Complex Plane
Problem 2ECP
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(a) Sketch the following set, M, in the complex plane: M = { EC || ≤ 1 and |zi| ≤ 1}. : (b) Find all solutions z E C of the equation z³ = 8i. You may leave your answers in polar or exponential form. (c) By writing the numbers 1+ i√√3 and 1 - i√√3 in polar or exponential form, determine all natural numbers n for which (1 + i√√3)" = (1 − i√√3)". 1+z (d) Consider complex numbers z, w such that = W. 1-2 i. Show that if zia where a is a real number, then w = 1. ii. Conversely, show that if w❘: =1 then has the form z = ia with a ЄR.
Transcribed Image Text:(a) Sketch the following set, M, in the complex plane: M = { EC || ≤ 1 and |zi| ≤ 1}. : (b) Find all solutions z E C of the equation z³ = 8i. You may leave your answers in polar or exponential form. (c) By writing the numbers 1+ i√√3 and 1 - i√√3 in polar or exponential form, determine all natural numbers n for which (1 + i√√3)" = (1 − i√√3)". 1+z (d) Consider complex numbers z, w such that = W. 1-2 i. Show that if zia where a is a real number, then w = 1. ii. Conversely, show that if w❘: =1 then has the form z = ia with a ЄR.
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