Explanation Given: The equation is z 6 − 1 = 0 . Formula Used : The polar form of a complex number is z = r ( cos θ + i sin θ ) Then the n distinct complex roots are ω k = r 1 n ( cos ( θ + 2 π k n ) + i sin ( θ + 2 π k n ) ) for k = 0 , 1 , 2 , 3... Calculation: The equation z 6 − 1 = 0 can be written as z 6 = 1 . So, evaluate sixth roots of 1 . The polar form of a complex number is, z = r ( cos θ + i sin θ ) Where, The number r is the modulus of z . r = | z | = a 2 + b 2 The θ is the argument of z . The value of r is, r = a 2 + b 2 Substitute 1 for a and 0 for b . r = ( 1 ) 2 + ( 0 ) 2 = 1 The value of r is 1 The value of θ is tan θ = b a . Substitute 1 for a and 0 for b . tan θ = 0 1 = 0 Possible value of θ is 0 . The complex number lies in first quadrant, so the value of θ is 0 . The polar form of the complex number z is, z = r ( cos θ + i sin θ ) Substitute 1 for r and 0 for θ . z = 1 ( cos ( 0 ) + i sin ( 0 ) ) The rule of n t h Roots of complex number is ω k = r 1 n ( cos ( θ + 2 π k n ) + i sin ( θ + 2 π k n ) ) . Substitute 1 for r and 0 for θ ,. 6 for n and 0 for k . ω ° = 1 1 6 ( cos 0 + 2 π ( 0 ) 6 + i sin 0 + 2 π ( 0 ) 6 ) = ( cos 0 + i sin 0 ) = 1 Substitute 1 for r and 0 for θ ,. 6 for n and 1 for k . ω 1 = 1 1 6 ( cos 0 + 2 π ( 1 ) 6 + i sin 0 + 2 π ( 1 ) 6 ) = ( cos 2 π 6 + i sin 2 π 6 ) = ( cos π 3 + i sin π 3 ) = 1 2 + 3 2 i Substitute 1 for r and 0 for θ ,

Precalculus: Mathematics for Calculus (Standalone Book)

7th Edition

ISBN: 9781305071759

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Brooks Cole

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Chapter 8.3, Problem 90E

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To Find: The roots of the equation z6−1=0 .

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Chapter 8.3, Problem 89E

Chapter 8.3, Problem 91E

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Chapter 8 Solutions

Precalculus: Mathematics for Calculus (Standalone Book)

Ch. 8.1 - We can describe the location of a point in the...Ch. 8.1 - Prob. 2E Ch. 8.1 - Yes or No? If No, give a reason. 3. Do the polar...Ch. 8.1 - Yes or No? If No, give a reason. 4. Do the...Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E 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 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Points in Polar Coordinates Determine which point...Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Polar Coordinates to Rectangular Coordinates A...Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E 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 - Prob. 37E 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 - Prob. 43E 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 - Prob. 49E 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 - Prob. 55E 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 - Prob. 61E 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 - Prob. 67E 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 - To plot points in polar coordinates, we use a grid...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 - Prob. 19E 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 - Prob. 43E 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 - Rectangular to Polar Sketch a graph of the...Ch. 8.2 - Prob. 60E Ch. 8.2 - Prob. 61E Ch. 8.2 - Prob. 62E Ch. 8.2 - Orbit of a Satellite Scientists and engineers...Ch. 8.2 - Prob. 64E Ch. 8.2 - Prob. 65E Ch. 8.2 - Prob. 66E Ch. 8.2 - Prob. 67E Ch. 8.3 - A complex number z = a + bi has two parts: a is...Ch. 8.3 - Let z = a + bi. (a) The modulus of z is r =...Ch. 8.3 - (a) The complex number z = 1 + i in polar form is...Ch. 8.3 - How many different nth roots does a nonzero...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - A Complex Number and Its Modulus Graph the complex...Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Graphing a Complex Number and Its Complex...Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Graphing Sets of Complex Numbers Sketch the set in...Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Graphing Sets of Complex Numbers Sketch the set in...Ch. 8.3 - Prob. 25E 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 - Prob. 31E 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 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Polar Form of Complex Numbers Write the complex...Ch. 8.3 - Prob. 41E Ch. 8.3 - Prob. 42E Ch. 8.3 - Prob. 43E Ch. 8.3 - Polar Form of Complex Numbers Write the complex...Ch. 8.3 - Prob. 45E Ch. 8.3 - Prob. 46E Ch. 8.3 - Prob. 47E Ch. 8.3 - Prob. 48E Ch. 8.3 - Prob. 49E 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 - Prob. 55E 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 - Prob. 61E 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 - Prob. 67E Ch. 8.3 - Prob. 68E Ch. 8.3 - Prob. 69E Ch. 8.3 - Prob. 70E Ch. 8.3 - Prob. 71E Ch. 8.3 - Prob. 72E Ch. 8.3 - Prob. 73E Ch. 8.3 - Prob. 74E Ch. 8.3 - Prob. 75E Ch. 8.3 - Prob. 76E Ch. 8.3 - Prob. 77E Ch. 8.3 - Roots of Complex Numbers Find the indicated roots,...Ch. 8.3 - Prob. 79E Ch. 8.3 - Prob. 80E Ch. 8.3 - Prob. 81E Ch. 8.3 - Prob. 82E Ch. 8.3 - Prob. 83E Ch. 8.3 - Prob. 84E Ch. 8.3 - Prob. 85E 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 - (a) The parametric equations x = f(t) and y = g(t)...Ch. 8.4 - (a) True or False? The same curve can be described...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 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Prob. 14E Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Prob. 17E Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Prob. 19E Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Sketching a Curve by Eliminating the Parameter A...Ch. 8.4 - Prob. 27E Ch. 8.4 - Circular Motion The position of an object in...Ch. 8.4 - Prob. 29E Ch. 8.4 - Prob. 30E Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Prob. 33E Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Parametric Equations for Curves Find parametric...Ch. 8.4 - Prob. 37E Ch. 8.4 - Path of a Projectile Referring to Exercise 37,...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 - Prob. 54E Ch. 8.4 - Prob. 55E Ch. 8.4 - Prolate Cycloid (a) In Exercise 55 if the point P...Ch. 8.4 - Parametric Equations of a Hyperbola Eliminate the...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 - Hypocycloid A circle C of radius b rolls on the...Ch. 8.4 - Prob. 64E Ch. 8.4 - Longbow Curve In the following figure, the circle...Ch. 8.4 - Prob. 66E Ch. 8.4 - Eliminating the Parameter Eliminate the parameter ...Ch. 8.4 - The Rotary Engine The Mazda RX-8 uses an...Ch. 8.4 - Prob. 69E Ch. 8.4 - DISCOVER WRITE: More Information in Parametric...Ch. 8.4 - Prob. 71E Ch. 8 - (a) Explain the polar coordinate system. (b) Graph...Ch. 8 - Prob. 2RCC Ch. 8 - Prob. 3RCC Ch. 8 - Prob. 4RCC Ch. 8 - Prob. 5RCC Ch. 8 - Prob. 6RCC Ch. 8 - Prob. 7RCC Ch. 8 - Prob. 8RCC Ch. 8 - Prob. 9RCC Ch. 8 - Prob. 1RE Ch. 8 - Prob. 2RE Ch. 8 - Prob. 3RE Ch. 8 - Prob. 4RE Ch. 8 - Prob. 5RE Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Rectangular Equations to Polar Equations (a)...Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE Ch. 8 - Polar Equations to Rectangular Equations (a)...Ch. 8 - Prob. 18RE Ch. 8 - Prob. 19RE Ch. 8 - Prob. 20RE Ch. 8 - Prob. 21RE Ch. 8 - Prob. 22RE Ch. 8 - Prob. 23RE Ch. 8 - Prob. 24RE Ch. 8 - Prob. 25RE Ch. 8 - Prob. 26RE Ch. 8 - Prob. 27RE Ch. 8 - Prob. 28RE Ch. 8 - Prob. 29RE Ch. 8 - Complex Numbers A complex number is given. (a)...Ch. 8 - Prob. 31RE Ch. 8 - Prob. 32RE Ch. 8 - Prob. 33RE Ch. 8 - Prob. 34RE Ch. 8 - Prob. 35RE Ch. 8 - Powers Using De Moivres Theorem Use De Moivres...Ch. 8 - Prob. 37RE Ch. 8 - Prob. 38RE Ch. 8 - Prob. 39RE Ch. 8 - Roots of Complex Numbers Find the indicated roots....Ch. 8 - Prob. 41RE Ch. 8 - Prob. 42RE Ch. 8 - Parametric Curves A pair of parametric equations...Ch. 8 - Prob. 44RE Ch. 8 - Prob. 45RE Ch. 8 - Prob. 46RE Ch. 8 - Prob. 47RE Ch. 8 - Prob. 48RE Ch. 8 - Finding Parametric Equations for a Curve In the...Ch. 8 - Prob. 1T Ch. 8 - Prob. 2T Ch. 8 - Prob. 3T Ch. 8 - Prob. 4T Ch. 8 - Prob. 5T Ch. 8 - Prob. 6T Ch. 8 - (a) Sketch the curve represented by the parametric...Ch. 8 - Find parametric equations for the line of slope 2...Ch. 8 - The position of an object in circular motion is...Ch. 8 - Prob. 1P Ch. 8 - Path of a Baseball Suppose a baseball is thrown at...Ch. 8 - Path of a Rocket Suppose that a rocket is fired at...Ch. 8 - Firing a Missile The initial speed of a missile is...Ch. 8 - Prob. 5P Ch. 8 - Shooting into the Wind Suppose that a projectile...Ch. 8 - Prob. 7P

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The roots of the equation z 6 − 1 = 0 .

Solution Summary: The author evaluates the roots of the equation z6-1=0.

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Chapter 8 Solutions