Question

and Demoiver's Theore Chapter a Section 9.6- Complex numbers Problem: Find polar and Cartesian forms for all three cube roots of the complex number -8. answers enter the polar form in in increasing onder of their angles, with OLO Į ait. Let Pq , P2, P3 represent the polar form and C., C2, C3 represent the corresponding Cartesian form. Round the constants three decimal places, if required. P. - C. = P2 = Il a Pz: C3

Answer 1

the answers can be summarised as:

1. Answers in polar forms are:

P1 = 2 cos 60 + sin 60

P2 = 2cos 180 + sin 180

P3 = 2 cos 300 + sin 300

2. Answers in the cartesian form are:

cl = 1+1.7321

C = -2 +02

C3 = 1 - 1,732

to Step : Our aim is polar and cartesian forms for all three cube roots of complex number of since, we Know every real number is complex number is not. but reverse is complex number, so, z = - 8 Therefore, express into Now, standard complex term zauty z = - 8toi x=-8f y=0

z=-8+01 into polar form stepe: Express the complex number z= rccoco tising) L Here, x²+4= o = tant (s) for x >0 tan ( 4 ) + 180 Cor) xco from ea put value x & y to find rto r= 6-8)2 +0² ву +180° tano+ 180° o= tant (2) o=0° +180° -> 0= 180 z = be written as: Hence, 8 (Ls 180° + i sin 180° generalized polas forman can z= 8hos (360°k +180) + i sin (360°k +1803) Step : Apply De-Moivre's therrem to find cuberort that, De-Moirre's theorem to generalized equation finding sth rook. know Zx = 2% z pyn [ cos (300k +O) + i sin (300k to) where, k=0, 1, 2,

ملا cube roof have to find also, n = 3 0 = 180° 8=8 Zk = 1898 [ Los (360° K +1803 isin (360°k boºk +180 41900)] Zk = 2[ cos(120k + 60°) + i sin (120k +60%)] Now K=0 , get first root put to = 2 [cos (609 tisin (60) = 2[ Ř tive] Zo = 1+ 3 i lly z,= 2 [cos (120°*1+60°) tisin (128x1 +60°)] 2 [cos 180° + isin 1800] z = z, = 2[++ixo] => 2=-2t0i + 2z = 2 [cos (120*2+609 + isin (120 A2+60)] 2₂ = 2 [ cos (300° + isin (300)] 22=2[ £ +i (-3) >

the answers as required. step 6: Rewrite polar form of soots are : P, = 2[cos 60+ i sin 60% P2 = 2 [cosisoº + isin 1800] P2 = 2 l cos 380° + i sin 300"] and cartesian form of soots are : B 1-107321 C₂ = -2 toi i-di- 1-1.7321