To determine The two square roots of the complex number − 5 + 12 j by using De Moivre’s theorem. 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 De Moivre’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 complex number is − 5 + 12 j . Compute the polar form of the complex number − 5 + 12 j as follows. Compare the complex number − 5 + 12 j with the general form of the complex number x + y j . Here, x = − 5 and y = 12 . Obtain the value of r as follows. Substitute x = − 5 and y = 12 in r 2 = x 2 + y 2 . r 2 = ( − 5 ) 2 + 12 2 r = ( − 5 ) 2 + 12 2 r = 25 + 144 r = 13 That is, the value of r is 13. Obtain the angle θ as follows. Substitute x = − 5 and y = 12 in tan θ = y x . tan θ = ( − 12 5 ) θ ref = tan − 1 ( − 12 5 ) θ ref = − 67

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 12.6, Problem 38E

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The two square roots of the complex number −5+12j by using De Moivre’s theorem.

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Chapter 12.6, Problem 37E

Chapter 12.6, Problem 39E

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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 - 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