Explanation Given: The complex number is 4 3 + 4 i . Approach: The polar form of a complex number z = x + i y is z = r ( cos θ + i sin θ ) , Here, r = x 2 + y 2 is the modulus of z and θ = tan − 1 ( y x ) is an argument of z . To find the n th roots of z = r ( cos θ + i sin θ ) , first find a complex number w such that w n = z , then the n th roots of z are w k = r 1 / n ( cos ( θ + 2 π k n ) + i sin ( θ + 2 π k n ) ) … ( 1 ) For k = 0 , 1 , 2 , … , n − 1 . Here, n is a positive number. Calculation: For the complex number z = 4 3 + 4 i the values of modulus r and the argument θ determined is as follows. r = x 2 + y 2 = ( 4 3 ) 2 + ( 4 ) 2 = 64 = 8 θ = tan − 1 ( 4 4 3 ) = tan − 1 ( 1 3 ) = π 6 Thus, the polar form for z is 8 ( cos π 6 + i sin π 6 ) To find square root, substitute 3 for n , 8 for r and π 6 for θ in equation ( 1 ) . w k = 8 1 / 3 ( cos ( π / 6 + 2 π k 3 ) + i sin ( π / 6 + 2 π k 3 ) ) = 2 ( cos ( π 18 + 2 π k 3 ) + i sin ( π 18 + 2 π k 3 ) ) … ( 2 ) The values of k will lies from 0 to n − 1 . The value of n is 3 , thus values of k will go from 0 to 2 . Substitute 0 for k in equation ( 2 ) w 0 = 2 ( cos ( π 18 + 2 π ( 0 ) 3 ) + i sin ( π 18 + 2 π ( 0 ) 3 ) ) = 2 ( cos ( π 18 ) + i sin ( π 18 ) ) Substitute 1 for k in equation ( 2 )

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN: 9781305071742

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 8.3, Problem 78E

To determine

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The cube roots of 43+4i and graph of the roots in complex plane.

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Chapter 8.3, Problem 77E

Chapter 8.3, Problem 79E

Chapter 8 Solutions

Algebra and Trigonometry (MindTap Course List)

Ch. 8.1 - CONCEPTS We can describe the location of a point...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - SKILLS 5-10 Plotting Points in Polar Coordinates...Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - SKILLS 11-16 Different Polar Coordinates for the...Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - SKILLS 17-24 Points in Polar Coordinates...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - SKILLS 25-26 Rectangular Coordinates to Polar...Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - 29-36 Polar Coordinates to Rectangular Coordinates...Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - SKILLS 37-44 Rectangular Coordinates to Polar...Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - 37-44 Rectangular Coordinates to Polar Coordinates...Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - SKILLS 45-50 Rectangular equations to polar...Ch. 8.1 - Prob. 50E Ch. 8.1 - Prob. 51E Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - SKILLS 51-70 Polar Equations to Rectangular...Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - Prob. 58E Ch. 8.1 - Prob. 59E Ch. 8.1 - Prob. 60E Ch. 8.1 - SKILLS 51-70 Polar Equations to Rectangular...Ch. 8.1 - Prob. 62E Ch. 8.1 - Prob. 63E Ch. 8.1 - Prob. 64E Ch. 8.1 - Prob. 65E Ch. 8.1 - Prob. 66E Ch. 8.1 - SKILLS 51-70 Polar Equations to Rectangular...Ch. 8.1 - Prob. 68E Ch. 8.1 - Prob. 69E Ch. 8.1 - Prob. 70E Ch. 8.1 - Prob. 71E Ch. 8.1 - Prob. 72E Ch. 8.2 - Prob. 1E Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - 17-22 Polar to Rectangular Sketch a graph of the...Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - Prob. 41E Ch. 8.2 - Prob. 42E Ch. 8.2 - 2346 Graphing Polar EquationsSketch a graph of the...Ch. 8.2 - Prob. 44E Ch. 8.2 - Prob. 45E Ch. 8.2 - Prob. 46E Ch. 8.2 - Prob. 47E Ch. 8.2 - Prob. 48E Ch. 8.2 - Prob. 49E Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Prob. 52E Ch. 8.2 - Prob. 53E Ch. 8.2 - Prob. 54E Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.2 - Prob. 57E Ch. 8.2 - Prob. 58E Ch. 8.2 - Prob. 59E Ch. 8.2 - Prob. 60E Ch. 8.2 - Prob. 61E Ch. 8.2 - Prob. 62E Ch. 8.2 - Prob. 63E Ch. 8.2 - Prob. 64E Ch. 8.2 - Prob. 65E Ch. 8.2 - Prob. 66E Ch. 8.2 - DISCUSSDISCOVERPROVEWRITE DISCUSS: Choosing a...Ch. 8.3 - CONCEPTS A complex number z=a+bi has two parts: a...Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - SKILLS 514 A Complex Number and Its Modulus Graph...Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - SKILLS 514A Complex Number and Its Modulus Graph...Ch. 8.3 - Prob. 14E Ch. 8.3 - SKILLS 15-16Graphing Complex Numbers. Sketch the...Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - SKILLS 19-20Graphing Complex Numbers. Sketch the...Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - 21-28 Graphing Sets of Complex Numbers Sketch the...Ch. 8.3 - 21-28 Graphing Sets of Complex Numbers Sketch the...Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - 2948 Polar Form of Complex Numbers Write the...Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8.3 - 2948 Polar Form of Complex Numbers Write the...Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Prob. 40E Ch. 8.3 - Prob. 41E Ch. 8.3 - Prob. 42E Ch. 8.3 - 2948 Polar Form of Complex Numbers Write the...Ch. 8.3 - Prob. 44E Ch. 8.3 - Prob. 45E Ch. 8.3 - Prob. 46E Ch. 8.3 - Prob. 47E Ch. 8.3 - Prob. 48E Ch. 8.3 - SKILLS 49-56Product and Quotients of Complex...Ch. 8.3 - Prob. 50E Ch. 8.3 - Prob. 51E Ch. 8.3 - Prob. 52E Ch. 8.3 - Prob. 53E Ch. 8.3 - Prob. 54E Ch. 8.3 - 49-56 Product and Quotients of Complex numbersFind...Ch. 8.3 - Prob. 56E Ch. 8.3 - Prob. 57E Ch. 8.3 - Prob. 58E Ch. 8.3 - Prob. 59E Ch. 8.3 - Prob. 60E Ch. 8.3 - 57-64 Product and Quotients of Complex...Ch. 8.3 - Prob. 62E Ch. 8.3 - Prob. 63E Ch. 8.3 - Prob. 64E Ch. 8.3 - Prob. 65E Ch. 8.3 - Prob. 66E Ch. 8.3 - SKILLS 65-76Powers Using De Moivres TheoremFind...Ch. 8.3 - SKILLS 65-76Powers Using De Moivres TheoremFind...Ch. 8.3 - Prob. 69E Ch. 8.3 - Prob. 70E Ch. 8.3 - Prob. 71E Ch. 8.3 - Prob. 72E Ch. 8.3 - SKILLS 65-76Powers Using De Moivres TheoremFind...Ch. 8.3 - Prob. 74E Ch. 8.3 - Prob. 75E Ch. 8.3 - Prob. 76E Ch. 8.3 - Prob. 77E Ch. 8.3 - Prob. 78E Ch. 8.3 - SKILLS 77-86Roots of Complex NumbersFind the...Ch. 8.3 - Prob. 80E Ch. 8.3 - Prob. 81E Ch. 8.3 - 77-86Roots of Complex NumbersFind the indicated...Ch. 8.3 - Prob. 83E Ch. 8.3 - Prob. 84E Ch. 8.3 - 77-86 Roots of Complex NumbersFind the indicated...Ch. 8.3 - Prob. 86E Ch. 8.3 - Prob. 87E Ch. 8.3 - Prob. 88E Ch. 8.3 - Prob. 89E Ch. 8.3 - Prob. 90E Ch. 8.3 - Prob. 91E Ch. 8.3 - Prob. 92E Ch. 8.3 - Prob. 93E Ch. 8.3 - Prob. 94E Ch. 8.3 - Prob. 95E Ch. 8.3 - Prob. 96E Ch. 8.3 - Prob. 97E Ch. 8.3 - Prob. 98E Ch. 8.3 - Prob. 99E Ch. 8.3 - Prob. 100E Ch. 8.3 - Prob. 101E Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8.4 - Prob. 27E Ch. 8.4 - Prob. 28E Ch. 8.4 - Prob. 29E Ch. 8.4 - Prob. 30E Ch. 8.4 - Prob. 31E Ch. 8.4 - Prob. 32E Ch. 8.4 - Prob. 33E Ch. 8.4 - Prob. 34E Ch. 8.4 - Prob. 35E Ch. 8.4 - Prob. 36E Ch. 8.4 - Prob. 37E Ch. 8.4 - Prob. 38E Ch. 8.4 - Prob. 39E Ch. 8.4 - Prob. 40E Ch. 8.4 - Prob. 41E Ch. 8.4 - Prob. 42E Ch. 8.4 - Prob. 43E Ch. 8.4 - Prob. 44E Ch. 8.4 - Prob. 45E Ch. 8.4 - Prob. 46E Ch. 8.4 - Prob. 47E Ch. 8.4 - Prob. 48E Ch. 8.4 - Prob. 49E Ch. 8.4 - Prob. 50E Ch. 8.4 - Prob. 51E Ch. 8.4 - Prob. 52E Ch. 8.4 - Prob. 53E Ch. 8.4 - Finding Parametric Equations for a Curve Two...Ch. 8.4 - Prob. 55E Ch. 8.4 - Prob. 56E Ch. 8.4 - Prob. 57E Ch. 8.4 - Prob. 58E Ch. 8.4 - Prob. 59E Ch. 8.4 - Prob. 60E Ch. 8.4 - Prob. 61E Ch. 8.4 - Prob. 62E Ch. 8.4 - Prob. 63E Ch. 8.4 - Epicycloid If the circle C of Exercise 63 rolls on...Ch. 8.4 - Longbow CurveIn the following figure, the circle...Ch. 8.4 - Prob. 66E Ch. 8.4 - Prob. 67E Ch. 8.4 - Prob. 68E Ch. 8.4 - Prob. 69E Ch. 8.4 - Prob. 70E Ch. 8.4 - Prob. 71E Ch. 8.CR - Prob. 1CC Ch. 8.CR - Prob. 2CC Ch. 8.CR - Prob. 3CC Ch. 8.CR - Prob. 4CC Ch. 8.CR - a How do we express the complex number z in polar...Ch. 8.CR - Prob. 6CC Ch. 8.CR - Prob. 7CC Ch. 8.CR - Prob. 8CC Ch. 8.CR - Prob. 9CC Ch. 8.CR - Prob. 1E Ch. 8.CR - Prob. 2E Ch. 8.CR - Prob. 3E Ch. 8.CR - Prob. 4E Ch. 8.CR - Prob. 5E Ch. 8.CR - Prob. 6E Ch. 8.CR - Prob. 7E Ch. 8.CR - Prob. 8E Ch. 8.CR - Prob. 9E Ch. 8.CR - Prob. 10E Ch. 8.CR - Prob. 11E Ch. 8.CR - Prob. 12E Ch. 8.CR - Prob. 13E Ch. 8.CR - Prob. 14E Ch. 8.CR - Prob. 15E Ch. 8.CR - Prob. 16E Ch. 8.CR - Prob. 17E Ch. 8.CR - Prob. 18E Ch. 8.CR - Prob. 19E Ch. 8.CR - Prob. 20E Ch. 8.CR - Prob. 21E Ch. 8.CR - Prob. 22E Ch. 8.CR - Prob. 23E Ch. 8.CR - Prob. 24E Ch. 8.CR - Prob. 25E Ch. 8.CR - Prob. 26E Ch. 8.CR - Prob. 27E Ch. 8.CR - Prob. 28E Ch. 8.CR - Prob. 29E Ch. 8.CR - Prob. 30E Ch. 8.CR - Prob. 31E Ch. 8.CR - Prob. 32E Ch. 8.CR - Prob. 33E Ch. 8.CR - Prob. 34E Ch. 8.CR - Prob. 35E Ch. 8.CR - Prob. 36E Ch. 8.CR - Prob. 37E Ch. 8.CR - Prob. 38E Ch. 8.CR - Prob. 39E Ch. 8.CR - Prob. 40E Ch. 8.CR - Prob. 41E Ch. 8.CR - Prob. 42E Ch. 8.CR - Prob. 43E Ch. 8.CR - Prob. 44E Ch. 8.CR - Prob. 45E Ch. 8.CR - Prob. 46E Ch. 8.CR - Prob. 47E Ch. 8.CR - Prob. 48E Ch. 8.CR - Prob. 49E Ch. 8.CT - Prob. 1CT Ch. 8.CT - Prob. 2CT Ch. 8.CT - Prob. 3CT Ch. 8.CT - Prob. 4CT Ch. 8.CT - Prob. 5CT Ch. 8.CT - Find the cube roots of 27i, and sketch these roots...Ch. 8.CT - Prob. 7CT Ch. 8.CT - Prob. 8CT Ch. 8.CT - Prob. 9CT Ch. 8.FOM - Trajectories Are Parabolas From the graphs in...Ch. 8.FOM - Path of a Baseball Suppose a baseball is thrown at...Ch. 8.FOM - Path of a Rocket Suppose that a rocket is fired at...Ch. 8.FOM - Firing a Missile The initial speed of a missile is...Ch. 8.FOM - Prob. 5P Ch. 8.FOM - Shooting into the Wind Suppose that a projectile...Ch. 8.FOM - Shooting into the Wind Using the parametric...Ch. 8.FOM - Prob. 8P

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