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The polar form of –256 – 256V3i is 512 (cos (-) + i sin (-)) (for steps, see polar form calculator). According to the De Moivre's Formula, all n-th roots of a complex number 0+2rk r (cos (0) + i sin (0)) are given by ri (cos (27k) + i sin (2k)), k = ++2Tk OS 0..n 1. We have that r = 512, 0 쏙, and n = 9. 3 + i sin (-30)) = 2 cos () - 2i sin () +2.T.0 k = 0: V512 ( cos 9. 2 (cos (-) + i sin (-)) • k = 1: V512 (cos (-1) 2 (cos () + i sin () = 2 cos () + 2i sin • k = 2: V512 (cos (=2) 27. +2-7-1 k = 1: V512 ( cos +i sin || 9. 9. 27 +2-7-2 (+2--2 + i sin 10T 10T 107 10T 2 (cos () + i sin ()) = 2 cos • k = 3: V512 (cos (=3) 2 (cos () + i sin ()) • k = 4: V512 (cos (=34) 2 (cos () +isin (주)) • k = 5: V512 (cos (-) 2 (cos () + i sin ()) = -2 cos () – 2i sin () • k = 6: V512 (cos (=3*) 2 (cos () + i sin ()) • k = 7: V512 (cos ( ) 2 (cos () + i sin ()) = -2 cos ( • k = 8: V512 (cos (s) + 2 (cos () + i sin (r )) = 2 cos ( + 2i sin 27 27 +i sin -좋+2·T:3 9. 16т 167 11T 11T = -2 cos () + 2i sin () + i sin (=4)) -2 cos () + 2i sin ( +i sin (-an) +2π.4 +2-7-4 9. 22т 227 27 +2-T-5 +2-T-5 9. 28T 287 +2-7-6) + i sin (=n)) -2 cos (E) – 2i sin (E) +2--7 +2-т-6 9. 34T 34т +i sin 9. 40T - 2i sin () 풍+2.T-8 40T 137 13T COS 27 i sin (-n)) = +2.T-8 9. 467 27)) 9 46T 87 27 2i sin ()

The polar form of –256 – 256V3i is 512 (cos (-) + i sin (-)) (for steps, see polar form calculator). According to the De Moivre's Formula, all n-th roots of a complex number 0+2rk r (cos (0) + i sin (0)) are given by ri (cos (27k) + i sin (2k)), k = ++2Tk OS 0..n 1. We have that r = 512, 0 쏙, and n = 9. 3 + i sin (-30)) = 2 cos () - 2i sin () +2.T.0 k = 0: V512 ( cos 9. 2 (cos (-) + i sin (-)) • k = 1: V512 (cos (-1) 2 (cos () + i sin () = 2 cos () + 2i sin • k = 2: V512 (cos (=2) 27. +2-7-1 k = 1: V512 ( cos +i sin || 9. 9. 27 +2-7-2 (+2--2 + i sin 10T 10T 107 10T 2 (cos () + i sin ()) = 2 cos • k = 3: V512 (cos (=3) 2 (cos () + i sin ()) • k = 4: V512 (cos (=34) 2 (cos () +isin (주)) • k = 5: V512 (cos (-) 2 (cos () + i sin ()) = -2 cos () – 2i sin () • k = 6: V512 (cos (=3*) 2 (cos () + i sin ()) • k = 7: V512 (cos ( ) 2 (cos () + i sin ()) = -2 cos ( • k = 8: V512 (cos (s) + 2 (cos () + i sin (r )) = 2 cos ( + 2i sin 27 27 +i sin -좋+2·T:3 9. 16т 167 11T 11T = -2 cos () + 2i sin () + i sin (=4)) -2 cos () + 2i sin ( +i sin (-an) +2π.4 +2-7-4 9. 22т 227 27 +2-T-5 +2-T-5 9. 28T 287 +2-7-6) + i sin (=n)) -2 cos (E) – 2i sin (E) +2--7 +2-т-6 9. 34T 34т +i sin 9. 40T - 2i sin () 풍+2.T-8 40T 137 13T COS 27 i sin (-n)) = +2.T-8 9. 467 27)) 9 46T 87 27 2i sin ()

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How to solve the other values (like 4pi/27 10pi/27) in detail if Wk = 2 cjs (-2/27+k 2pi/9)
The polar form of –256 – 256v/3i is 512 (cos (-) + i sin (-)) (for steps, see polar form calculator). According to the De Moivre's Formula, all n-th roots of a complex number 0+2™k r (cos (0) + i sin (0)) are given by ri (cos (27k) + i sin (2k)), k = ++2™k OS 0..n 1. We have that r = 512, 0 꼭, and n = 9. 3 + i sin (-30)) = 2 cos () – 2i sin () +2-7-0 k = 0: V512 ( cos 9. 2 (cos (-) + i sin (-)) • k = 1: V512 (cos (-1) 2 (cos () + i sin () = 2 cos () + 2i sin • k = 2: V512 (cos (=3*2) 27. +2-7-1 k = 1: V512 ( cos +i sin 9. 4T 9. 27 -풍 +2·T-2 +i sin (-+2--2 10t 10T 107 2 (cos () + i sin ())= 2 cos • k = 3: V512 (cos (=3) 2 (cos () + i sin () • k = 4: V512 (cos (=4) + 2 (cos (2) +isin (2주)) • k = 5: V512 (cos (-) + 2i sin (107 27 27 '-좋+2·T:3 +2•T-3 +i sin 9. 16т 9 11t 167 11T = -2 cos () + 2i sin () i sin (=4)) +2π.4 +2-7-4 9. 22т 227 -2 cos () + 2i sin ( +2-7-5 27 +i sin (n) +2-7-5 2 (cos () + i sin ()) = -2 cos () – 2i sin (5) -+2-7-6` + i sin (=n)) -2 cos (E) – 2i sin (E) +2-7-7 28т 287 • k = 6: V512 (cos (=3r) 2 (cos () + i sin ()) • k = 7: V512 (cos (=3?) +2-т-6 9. 34T 34T + i sin (-n7)) = 2 (cos () + i sin ()) = -2 cos ( - 2i sin () 40T 40T 137 13T COS 27 • k = 8: V512 (cos (s) + 2 (cos () + i sin (4r )) = 2 cos ( i sin (-n)) = +2.T-8 +2-7-8 9. 467 27)) 87 27 2i sin (7)
Transcribed Image Text:The polar form of –256 – 256v/3i is 512 (cos (-) + i sin (-)) (for steps, see polar form calculator). According to the De Moivre's Formula, all n-th roots of a complex number 0+2™k r (cos (0) + i sin (0)) are given by ri (cos (27k) + i sin (2k)), k = ++2™k OS 0..n 1. We have that r = 512, 0 꼭, and n = 9. 3 + i sin (-30)) = 2 cos () – 2i sin () +2-7-0 k = 0: V512 ( cos 9. 2 (cos (-) + i sin (-)) • k = 1: V512 (cos (-1) 2 (cos () + i sin () = 2 cos () + 2i sin • k = 2: V512 (cos (=3*2) 27. +2-7-1 k = 1: V512 ( cos +i sin 9. 4T 9. 27 -풍 +2·T-2 +i sin (-+2--2 10t 10T 107 2 (cos () + i sin ())= 2 cos • k = 3: V512 (cos (=3) 2 (cos () + i sin () • k = 4: V512 (cos (=4) + 2 (cos (2) +isin (2주)) • k = 5: V512 (cos (-) + 2i sin (107 27 27 '-좋+2·T:3 +2•T-3 +i sin 9. 16т 9 11t 167 11T = -2 cos () + 2i sin () i sin (=4)) +2π.4 +2-7-4 9. 22т 227 -2 cos () + 2i sin ( +2-7-5 27 +i sin (n) +2-7-5 2 (cos () + i sin ()) = -2 cos () – 2i sin (5) -+2-7-6` + i sin (=n)) -2 cos (E) – 2i sin (E) +2-7-7 28т 287 • k = 6: V512 (cos (=3r) 2 (cos () + i sin ()) • k = 7: V512 (cos (=3?) +2-т-6 9. 34T 34T + i sin (-n7)) = 2 (cos () + i sin ()) = -2 cos ( - 2i sin () 40T 40T 137 13T COS 27 • k = 8: V512 (cos (s) + 2 (cos () + i sin (4r )) = 2 cos ( i sin (-n)) = +2.T-8 +2-7-8 9. 467 27)) 87 27 2i sin (7)
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