Explanation Formula used: Polar form : For any point ( x , y ) in the complex plane, the equations relating x , y , r and θ are, x = r cos θ y = r sin θ r 2 = x 2 + y 2 tan θ = y x Where, the length r is the absolute value and θ is the argument of the complex number . The rectangular form x + y j of a complex number can be converted to the polar form as x + y j = r ( cos θ + j sin θ ) by using the above equations. Theorem used: Using DeMoivre’s Theorem to find the n nth roots of a complex number : “1. Express the number in polar form. 2. Express the root as a fractional exponent. 3. Use ( r ∠ θ ) n = r n ∠ n θ or [ r ( cos θ + j sin θ ) ] n = r n [ cos n θ + j sin n θ ] with θ to find one root. 4. Use ( r ∠ θ ) n = r n ∠ n θ or [ r ( cos θ + j sin θ ) ] n = r n [ cos n θ + j sin n θ ] and add 360 ° to θ , n − 1 times, to find the other roots. Calculation: The given equation is x 4 + j = 0 . Solve the given equation and compute the value of x as follows. x 4 + j = 0 x 4 = − j x = − j 4 Since the value of x is fourth root of – j , compute the four fourth roots of – j . Write the polar form of the complex number – j . Compute the polar form of the complex number – j as follows. Compare the complex number – j with the general form of the complex number x + y j . Here, x = 0 and y = − 1 . Obtain the value of r as follows. Substitute x = 0 and y = − 1 in r 2 = x 2 + y 2 . r 2 = 0 2 + ( − 1 ) 2 r = 0 2 + ( − 1 ) 2 r = 1 2 r = 1 That is, the value of r is 1. Obtain the angle θ as follows. Substitute x = 0 and y = − 1 in tan θ = y x . tan θ = − 1 0 θ ref = tan − 1 ( − 1 0 ) θ ref = 270 ° That is, the angle θ ref is 270 ° . Thus, the polar form of the complex number j is 1 ∠ 270 ° , that is, 1 ( cos 270 ° + j sin 270 ° ) . Compute four fourth roots of 1 ( cos 270 ° + j sin 270 ° ) as follows

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Author: Washington

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Chapter 12, Problem 71RE

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Chapter 12, Problem 70RE

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Chapter 12 Solutions

Basic Technical Mathematics

Ch. 12.1 - Write in terms of j. Ch. 12.1 - Simplify: 2. Ch. 12.1 - Simplify: 2. Ch. 12.1 - Prob. 4PE Ch. 12.1 - Prob. 5PE Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - In Exercises 1–4, perform the indicated operations...Ch. 12.1 - In Exercises 5–16, express each number in terms of...

Ch. 12.1 - In Exercises 5–16, express each number in terms of...Ch. 12.1 - In Exercises 5–16, express each number in terms of...Ch. 12.1 - Prob. 8E Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E Ch. 12.1 - Prob. 11E Ch. 12.1 - Prob. 12E Ch. 12.1 - Prob. 13E Ch. 12.1 - Prob. 14E Ch. 12.1 - Prob. 15E Ch. 12.1 - Prob. 16E Ch. 12.1 - In Exercises 17–32, simplify each of the given...Ch. 12.1 - Prob. 18E Ch. 12.1 - Prob. 19E Ch. 12.1 - Prob. 20E Ch. 12.1 - Prob. 21E Ch. 12.1 - Prob. 22E Ch. 12.1 - Prob. 23E Ch. 12.1 - Prob. 24E Ch. 12.1 - In Exercises 17–32, simplify each of the given...Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.1 - Prob. 28E Ch. 12.1 - Prob. 29E Ch. 12.1 - Prob. 30E Ch. 12.1 - Prob. 31E Ch. 12.1 - Prob. 32E Ch. 12.1 - Prob. 33E Ch. 12.1 - Prob. 34E Ch. 12.1 - Prob. 35E Ch. 12.1 - Prob. 36E Ch. 12.1 - Prob. 37E Ch. 12.1 - Prob. 38E Ch. 12.1 - Prob. 39E Ch. 12.1 - Prob. 40E Ch. 12.1 - Prob. 41E Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Prob. 45E Ch. 12.1 - Prob. 46E Ch. 12.1 - Prob. 47E Ch. 12.1 - Prob. 48E Ch. 12.1 - Prob. 49E Ch. 12.1 - In Exercises 33–50, perform the indicated...Ch. 12.1 - Prob. 51E Ch. 12.1 - Prob. 52E Ch. 12.1 - Prob. 53E Ch. 12.1 - Prob. 54E Ch. 12.1 - Prob. 55E Ch. 12.1 - Prob. 56E Ch. 12.1 - Prob. 57E Ch. 12.1 - Prob. 58E Ch. 12.1 - In Exercises 55–60, find the values of x and y...Ch. 12.1 - In Exercises 55–60, find the values of x and y...Ch. 12.1 - Prob. 61E Ch. 12.1 - Prob. 62E Ch. 12.1 - Prob. 63E Ch. 12.1 - Prob. 64E Ch. 12.1 - Prob. 65E Ch. 12.1 - Prob. 66E Ch. 12.1 - Prob. 67E Ch. 12.1 - Prob. 68E Ch. 12.1 - Prob. 69E Ch. 12.1 - Prob. 70E Ch. 12.1 - Prob. 71E Ch. 12.1 - Prob. 72E Ch. 12.1 - Prob. 73E Ch. 12.1 - Prob. 74E Ch. 12.2 - Prob. 1PE Ch. 12.2 - Prob. 2PE Ch. 12.2 - Prob. 3PE Ch. 12.2 - Prob. 1E Ch. 12.2 - Prob. 2E Ch. 12.2 - In Exercises 1-4, perform the indicated operations...Ch. 12.2 - Prob. 4E Ch. 12.2 - In Exercises 5–38, perform the indicated...Ch. 12.2 - Prob. 6E Ch. 12.2 - In Exercises 5–38, perform the indicated...Ch. 12.2 - Prob. 8E Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Prob. 11E Ch. 12.2 - In Exercises 5–38, perform the indicated...Ch. 12.2 - In Exercises 5–38, perform the indicated...Ch. 12.2 - Prob. 14E Ch. 12.2 - In Exercises 5–38, perform the indicated...Ch. 12.2 - Prob. 16E Ch. 12.2 - Prob. 17E Ch. 12.2 - In Exercises 5–38, perform the indicated...Ch. 12.2 - Prob. 19E Ch. 12.2 - Prob. 20E Ch. 12.2 - Prob. 21E Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Prob. 24E Ch. 12.2 - Prob. 25E Ch. 12.2 - Prob. 26E Ch. 12.2 - Prob. 27E Ch. 12.2 - Prob. 28E Ch. 12.2 - Prob. 29E Ch. 12.2 - Prob. 30E Ch. 12.2 - Prob. 31E Ch. 12.2 - Prob. 32E Ch. 12.2 - Prob. 33E Ch. 12.2 - Prob. 34E Ch. 12.2 - Prob. 35E Ch. 12.2 - Prob. 36E Ch. 12.2 - Prob. 37E Ch. 12.2 - Prob. 38E Ch. 12.2 - Prob. 39E Ch. 12.2 - Prob. 40E Ch. 12.2 - Prob. 41E Ch. 12.2 - Prob. 42E Ch. 12.2 - Prob. 43E Ch. 12.2 - Prob. 44E Ch. 12.2 - Prob. 45E Ch. 12.2 - Prob. 46E Ch. 12.2 - Prob. 47E Ch. 12.2 - Prob. 48E Ch. 12.2 - Prob. 49E Ch. 12.2 - In Exercises 43–56, solve the given...Ch. 12.2 - Prob. 51E Ch. 12.2 - Prob. 52E Ch. 12.2 - Prob. 53E Ch. 12.2 - Prob. 54E Ch. 12.2 - Prob. 55E Ch. 12.2 - Prob. 56E Ch. 12.2 - Prob. 57E Ch. 12.2 - Prob. 58E Ch. 12.2 - Prob. 59E Ch. 12.2 - Prob. 60E Ch. 12.2 - In Exercises 61-64, answer or explain as...Ch. 12.2 - Prob. 62E Ch. 12.2 - Prob. 63E Ch. 12.2 - Prob. 64E Ch. 12.3 - Prob. 1E Ch. 12.3 - Prob. 2E Ch. 12.3 - Prob. 3E Ch. 12.3 - Prob. 4E Ch. 12.3 - Prob. 5E Ch. 12.3 - Prob. 6E Ch. 12.3 - Prob. 7E Ch. 12.3 - Prob. 8E Ch. 12.3 - Prob. 9E Ch. 12.3 - Prob. 10E Ch. 12.3 - Prob. 11E Ch. 12.3 - Prob. 12E Ch. 12.3 - Prob. 13E Ch. 12.3 - Prob. 14E Ch. 12.3 - Prob. 15E Ch. 12.3 - Prob. 16E Ch. 12.3 - Prob. 17E Ch. 12.3 - Prob. 18E Ch. 12.3 - Prob. 19E Ch. 12.3 - Prob. 20E Ch. 12.3 - Prob. 21E Ch. 12.3 - Prob. 22E Ch. 12.3 - Prob. 23E Ch. 12.3 - Prob. 24E Ch. 12.3 - Prob. 25E Ch. 12.3 - Prob. 26E Ch. 12.3 - Prob. 27E Ch. 12.3 - Prob. 28E Ch. 12.3 - Prob. 29E Ch. 12.3 - Prob. 30E Ch. 12.3 - Prob. 31E Ch. 12.3 - Prob. 32E Ch. 12.3 - Prob. 33E Ch. 12.3 - Prob. 34E Ch. 12.3 - Prob. 35E Ch. 12.3 - Prob. 36E Ch. 12.3 - Prob. 37E Ch. 12.3 - Prob. 38E Ch. 12.4 - Prob. 1PE Ch. 12.4 - Prob. 2PE Ch. 12.4 - Prob. 3PE Ch. 12.4 - Prob. 1E Ch. 12.4 - In Exercises 1 and 2, change the sign of the real...Ch. 12.4 - Prob. 3E Ch. 12.4 - Prob. 4E Ch. 12.4 - Prob. 5E Ch. 12.4 - Prob. 6E Ch. 12.4 - Prob. 7E Ch. 12.4 - In Exercises 3-18, represent each complex number...Ch. 12.4 - Prob. 9E Ch. 12.4 - Prob. 10E Ch. 12.4 - Prob. 11E Ch. 12.4 - Prob. 12E Ch. 12.4 - Prob. 13E Ch. 12.4 - Prob. 14E Ch. 12.4 - Prob. 15E Ch. 12.4 - Prob. 16E Ch. 12.4 - Prob. 17E Ch. 12.4 - Prob. 18E Ch. 12.4 - In Exercises 19-36, represent each complex number...Ch. 12.4 - Prob. 20E Ch. 12.4 - Prob. 21E Ch. 12.4 - Prob. 22E Ch. 12.4 - Prob. 23E Ch. 12.4 - Prob. 24E Ch. 12.4 - Prob. 25E Ch. 12.4 - Prob. 26E Ch. 12.4 - Prob. 27E Ch. 12.4 - Prob. 28E Ch. 12.4 - In Exercises 19-36, represent each complex number...Ch. 12.4 - Prob. 30E Ch. 12.4 - Prob. 31E Ch. 12.4 - Prob. 32E Ch. 12.4 - Prob. 33E Ch. 12.4 - Prob. 34E Ch. 12.4 - In Exercises 19-36, represent each complex number...Ch. 12.4 - Prob. 36E Ch. 12.4 - Prob. 37E Ch. 12.4 - Prob. 38E Ch. 12.4 - Prob. 39E Ch. 12.4 - Prob. 40E Ch. 12.4 - In Exercises 37–44, solve the given problems. 41....Ch. 12.4 - In Exercises 37–44, solve the given problems. 42....Ch. 12.4 - Prob. 43E Ch. 12.4 - Prob. 44E Ch. 12.5 - Prob. 1PE Ch. 12.5 - Prob. 2PE Ch. 12.5 - Represent 3.00e2.66j in rectangular form. Ch. 12.5 - Prob. 1E Ch. 12.5 - Prob. 2E Ch. 12.5 - In Exercises 3-22, express the given numbers in...Ch. 12.5 - In Exercises 3-22, express the given numbers in...Ch. 12.5 - Prob. 5E Ch. 12.5 - Prob. 6E Ch. 12.5 - Prob. 7E Ch. 12.5 - Prob. 8E Ch. 12.5 - In Exercises 3-22, express the given numbers in...Ch. 12.5 - Prob. 10E Ch. 12.5 - Prob. 11E Ch. 12.5 - Prob. 12E Ch. 12.5 - Prob. 13E Ch. 12.5 - Prob. 14E Ch. 12.5 - In Exercises 3-22, express the given numbers in...Ch. 12.5 - Prob. 16E Ch. 12.5 - Prob. 17E Ch. 12.5 - Prob. 18E Ch. 12.5 - Prob. 19E Ch. 12.5 - Prob. 20E Ch. 12.5 - Prob. 21E Ch. 12.5 - Prob. 22E Ch. 12.5 - Prob. 23E Ch. 12.5 - In Exercises 23–30, express the given complex...Ch. 12.5 - Prob. 25E Ch. 12.5 - Prob. 26E Ch. 12.5 - In Exercises 23–30, express the given complex...Ch. 12.5 - Prob. 28E Ch. 12.5 - Prob. 29E Ch. 12.5 - Prob. 30E Ch. 12.5 - In Exercises 31–34, perform the indicated...Ch. 12.5 - Prob. 32E Ch. 12.5 - Prob. 33E Ch. 12.5 - Prob. 34E Ch. 12.5 - Prob. 35E Ch. 12.5 - Prob. 36E Ch. 12.5 - In Exercises 35–40, perform the indicated...Ch. 12.5 - Prob. 38E Ch. 12.5 - Prob. 39E Ch. 12.5 - In Exercises 35–40, perform the indicated...Ch. 12.6 - Prob. 1PE Ch. 12.6 - Prob. 2PE Ch. 12.6 - Find the polar form power: (3 cos 50°)8 Ch. 12.6 - Prob. 4PE Ch. 12.6 - Prob. 1E Ch. 12.6 - Prob. 2E Ch. 12.6 - Prob. 3E Ch. 12.6 - Prob. 4E Ch. 12.6 - Prob. 5E Ch. 12.6 - Prob. 6E Ch. 12.6 - Prob. 7E Ch. 12.6 - Prob. 8E Ch. 12.6 - Prob. 9E Ch. 12.6 - Prob. 10E Ch. 12.6 - Prob. 11E Ch. 12.6 - Prob. 12E Ch. 12.6 - Prob. 13E Ch. 12.6 - Prob. 14E Ch. 12.6 - Prob. 15E Ch. 12.6 - Prob. 16E Ch. 12.6 - Prob. 17E Ch. 12.6 - Prob. 18E Ch. 12.6 - Prob. 19E Ch. 12.6 - Prob. 20E Ch. 12.6 - Prob. 21E Ch. 12.6 - Prob. 22E Ch. 12.6 - Prob. 23E Ch. 12.6 - Prob. 24E Ch. 12.6 - Prob. 25E Ch. 12.6 - Prob. 26E Ch. 12.6 - Prob. 27E Ch. 12.6 - Prob. 28E Ch. 12.6 - Prob. 29E Ch. 12.6 - Prob. 30E Ch. 12.6 - Prob. 31E Ch. 12.6 - Prob. 32E Ch. 12.6 - Prob. 33E Ch. 12.6 - Prob. 34E Ch. 12.6 - Prob. 35E Ch. 12.6 - Prob. 36E Ch. 12.6 - Prob. 37E Ch. 12.6 - In Exercises 35–40, use DeMoivre’s theorem to find...Ch. 12.6 - Prob. 39E Ch. 12.6 - Prob. 40E Ch. 12.6 - Prob. 41E Ch. 12.6 - Prob. 42E Ch. 12.6 - Prob. 43E Ch. 12.6 - Prob. 44E Ch. 12.6 - In Exercises 41–46, find all of the roots of the...Ch. 12.6 - Prob. 46E Ch. 12.6 - Prob. 47E Ch. 12.6 - Prob. 48E Ch. 12.6 - Prob. 49E Ch. 12.6 - Prob. 50E Ch. 12.6 - Prob. 51E Ch. 12.6 - Prob. 52E Ch. 12.6 - The electric power p (in W) supplied to an element...Ch. 12.6 - Prob. 54E Ch. 12.6 - Prob. 55E Ch. 12.6 - Prob. 56E Ch. 12.7 - Prob. 1PE Ch. 12.7 - Prob. 1E Ch. 12.7 - Prob. 2E Ch. 12.7 - Prob. 3E Ch. 12.7 - Prob. 4E Ch. 12.7 - Prob. 5E Ch. 12.7 - Prob. 6E Ch. 12.7 - Prob. 7E Ch. 12.7 - Prob. 8E Ch. 12.7 - Prob. 9E Ch. 12.7 - Prob. 10E Ch. 12.7 - Prob. 11E Ch. 12.7 - Prob. 12E Ch. 12.7 - Prob. 13E Ch. 12.7 - Prob. 14E Ch. 12.7 - Prob. 15E Ch. 12.7 - Prob. 16E Ch. 12.7 - Prob. 17E Ch. 12.7 - Prob. 18E Ch. 12.7 - Prob. 19E Ch. 12.7 - Prob. 20E Ch. 12.7 - Prob. 21E Ch. 12.7 - Prob. 22E Ch. 12.7 - Prob. 23E Ch. 12.7 - Prob. 24E Ch. 12 - Prob. 1RE Ch. 12 - Prob. 2RE Ch. 12 - Prob. 3RE Ch. 12 - Prob. 4RE Ch. 12 - Prob. 5RE Ch. 12 - Prob. 6RE Ch. 12 - Prob. 7RE Ch. 12 - Prob. 8RE Ch. 12 - Prob. 9RE Ch. 12 - Prob. 10RE Ch. 12 - Prob. 11RE Ch. 12 - Prob. 12RE Ch. 12 - Prob. 13RE Ch. 12 - Prob. 14RE Ch. 12 - Prob. 15RE Ch. 12 - Prob. 16RE Ch. 12 - Prob. 17RE Ch. 12 - Prob. 18RE Ch. 12 - Prob. 19RE Ch. 12 - Prob. 20RE Ch. 12 - Prob. 21RE Ch. 12 - Prob. 22RE Ch. 12 - Prob. 23RE Ch. 12 - Prob. 24RE Ch. 12 - Prob. 25RE Ch. 12 - Prob. 26RE Ch. 12 - Prob. 27RE Ch. 12 - Prob. 28RE Ch. 12 - Prob. 29RE Ch. 12 - Prob. 30RE Ch. 12 - Prob. 31RE Ch. 12 - In Exercises 29–36, give the polar and exponential...Ch. 12 - In Exercises 29–36, give the polar and exponential...Ch. 12 - In Exercises 29–36, give the polar and exponential...Ch. 12 - In Exercises 29–36, give the polar and exponential...Ch. 12 - Prob. 36RE Ch. 12 - Prob. 37RE Ch. 12 - In Exercises 37–48, give the rectangular form of...Ch. 12 - In Exercises 37–48, give the rectangular form of...Ch. 12 - In Exercises 37–48, give the rectangular form of...Ch. 12 - In Exercises 37–48, give the rectangular form of...Ch. 12 - Prob. 42RE Ch. 12 - Prob. 43RE Ch. 12 - Prob. 44RE Ch. 12 - In Exercises 37–48, give the rectangular form of...Ch. 12 - Prob. 46RE Ch. 12 - Prob. 47RE Ch. 12 - Prob. 48RE Ch. 12 - Prob. 49RE Ch. 12 - Prob. 50RE Ch. 12 - Prob. 51RE Ch. 12 - Prob. 52RE Ch. 12 - Prob. 53RE Ch. 12 - Prob. 54RE Ch. 12 - Prob. 55RE Ch. 12 - Prob. 56RE Ch. 12 - Prob. 57RE Ch. 12 - Prob. 58RE Ch. 12 - Prob. 59RE Ch. 12 - Prob. 60RE Ch. 12 - Prob. 61RE Ch. 12 - Prob. 62RE Ch. 12 - Prob. 63RE Ch. 12 - Prob. 64RE Ch. 12 - Prob. 65RE Ch. 12 - Prob. 66RE Ch. 12 - Prob. 67RE Ch. 12 - Prob. 68RE Ch. 12 - Prob. 69RE Ch. 12 - Prob. 70RE Ch. 12 - Prob. 71RE Ch. 12 - Prob. 72RE Ch. 12 - Prob. 73RE Ch. 12 - Prob. 74RE Ch. 12 - Prob. 75RE Ch. 12 - Prob. 76RE Ch. 12 - Prob. 77RE Ch. 12 - Prob. 78RE Ch. 12 - Prob. 79RE Ch. 12 - Prob. 80RE Ch. 12 - Prob. 81RE Ch. 12 - Prob. 82RE Ch. 12 - Prob. 85RE Ch. 12 - Prob. 86RE Ch. 12 - Prob. 87RE Ch. 12 - Prob. 88RE Ch. 12 - Prob. 89RE Ch. 12 - Prob. 90RE Ch. 12 - Prob. 91RE Ch. 12 - Prob. 92RE Ch. 12 - Prob. 93RE Ch. 12 - Prob. 94RE Ch. 12 - Prob. 95RE Ch. 12 - Prob. 96RE Ch. 12 - Prob. 97RE Ch. 12 - Prob. 98RE Ch. 12 - Prob. 99RE Ch. 12 - Prob. 100RE Ch. 12 - Prob. 1PT Ch. 12 - Multiply, expressing the result in polar...Ch. 12 - Prob. 3PT Ch. 12 - Prob. 4PT Ch. 12 - Prob. 5PT Ch. 12 - Prob. 6PT Ch. 12 - Express 2.56(cos 125.2° + j sin 125.2°) in...Ch. 12 - Prob. 8PT Ch. 12 - Express 3.47 − 2.81j in exponential form. 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