Explanation Given information: z = 1 Concept Involved: Every real (or complex) number has exactly n distinct nth roots, a surprising and attractive fact about numbers. The key to finding these roots is trigonometric form for complex numbers. Suppose that z and w are complex numbers such that w is an nth root of z. That is, w = z n If we raise both sides of this last equation to nth power, we have w n = z . Now suppose z = r ( cos θ + i sin θ ) and z = s ( cos α + i sin α ) . Substituting these expressions into the equation w n = z , we have [ s ( cos α + i sin α ) ] n = r ( cos θ + i sin θ ) We can rewrite the left side of this last equation de Moivre’s Theorem. s n ( cos n α + i sin n α ) = r ( cos θ + i sin θ ) The only way these two expressions can be equal is if their moduli are equal and their arguments are coterminal. Moduli Arguments Equal Coterminal s n = r n α = θ + 360 ° k k is any integer Solving for s and α , we have s = r 1 / n α = θ + 360 ° k n = θ n + 360 ° n k To summarize, we find the nth roots of a complex number by first finding the real nth root of the modulus. The angle for the first root will be θ / n , and the angles for the remaining roots are found by adding multiples of 360 ° / n . The values of k will range from 0 to n - 1, because when k = n we would be adding 360 ° n ⋅ n = 360 ° which would be coterminal with the first angle. Theorem (Roots) The nth roots of the complex number z = r ( cos θ + i sin θ ) = r c i s θ are given by w k = r 1 / n [ cos ( θ n + 360 ° n k ) + i sin ( θ n + 360 ° n k ) ] = r 1 / n c i s ( θ n + 360 ° n k ) where k = 0 , 1 , 2 , ... , n − 1. The root w 0 , corresponding to k = 0 , is called the principal nth root of z. Calculation: TABLE 1 Steps Description z = 1 Complex number in Standard Form x + y i ⇒ 1 x = 1 y = 0 Identifying x and y by comparison r = x 2 + y 2 = 1 2 + 0 2 = 1 + 0 = 1 Identifying x and y cos θ = x r = 1 1 = 1 sin θ = y r = 0 1 = 0 Identifying cos θ and sin θ

8th Edition

ISBN: 8220101473318

Author: Mckeague

Publisher: CENGAGE L

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 8.4, Problem 36PS

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The six sixth roots for given complex number, leave the result in trigonometric form and graph all six roots on the same coordinate system

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Chapter 8 Solutions

EBK TRIGONOMETRY

Ch. 8.1 - For Questions 1 through 10, fill in the blank with...Ch. 8.1 - Prob. 2PS Ch. 8.1 - Prob. 3PS Ch. 8.1 - Prob. 4PS Ch. 8.1 - Prob. 5PS Ch. 8.1 - Prob. 6PS Ch. 8.1 - Prob. 7PS Ch. 8.1 - Prob. 8PS Ch. 8.1 - Prob. 9PS Ch. 8.1 - Prob. 10PS

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The six sixth roots for given complex number , leave the result in trigonometric form and graph all six roots on the same coordinate system

Solution Summary: The author explains how to find the six sixth roots for a given complex number using trigonometric form.

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To find:

The six sixth roots for given complex number, leave the result in trigonometric form and graph all six roots on the same coordinate system

Chapter 8 Solutions

The six sixth roots for given complex number , leave the result in trigonometric form and graph all six roots on the same coordinate system

Solution Summary: The author explains how to find the six sixth roots for a given complex number using trigonometric form.

Videos

To find: The six sixth roots for given complex number, leave the result in trigonometric form and graph all six roots on the same coordinate system

Chapter 8 Solutions

To find:

The six sixth roots for given complex number, leave the result in trigonometric form and graph all six roots on the same coordinate system