(a) Given a complex number u =2+ 3i.(i) Determine z = u? + 13 – 4i in the form a + ib.(ii) Express z in polar form.(iii) Solve w3= z and sketch the roots on a single Argand Diagram.(b) Use de Moivre's theorem to show thatsin 30 = 3 sin 0 – 4 sin3 0.Hence, obtain all solutions of x for the following equation:4.x – 3x + 1 = 0.

Answered: (a) Given a complex number u = 2+ 3i.…

(a) Given a complex number u = 2+ 3i. (i) Determine z = u? + 13 – 4i in the form a + ib. %3D (ii) Express z in polar form. (iii) Solve w3 = z and sketch the roots on a single Argand Diagram. %3D

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter8: Polar Coordinates And Parametric Equations

Section8.3: Polar Form Of Complex Numbers; De Moivre's Theorem

Problem 3E

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Roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.

Question

(a) Given a complex number u =
2+ 3i.
(i) Determine z = u? + 13 – 4i in the form a + ib.
(ii) Express z in polar form.
(iii) Solve w3
= z and sketch the roots on a single Argand Diagram.
(b) Use de Moivre's theorem to show that
sin 30 = 3 sin 0 – 4 sin3 0.
Hence, obtain all solutions of x for the following equation:
4.x – 3x + 1 = 0.

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