Question

Find all solutions of 24 = 4 + 4i Find all solutions of z2 = 3 - 2i

Answer 1

(1) Z"=4+ui z = a + atbi = izl(cos(o)+ i since)) (Z1 = Ja²+br 2 = arbi Iz = N444 = 2 jy Os to tan' (44) - z = UJI (cos( t) + i sin(*)) De Moivre's theorem (cosche) + sin(4e)) = 4+4 = 4 JZ (cos(6)+ i sin (5)) t" cu → r= $2,- 2 and cos (40) = COS (2670) and Sin (46) = sincerin) 0:0, ar r=0. z = y tui 2p = 1.414 (cos (0) + i sin cos). 20= 1. Gry aut r=1" 40 = 20 () o = 1/2 1.414 (cos ( ID+isin (11) 1.414 i ti

at r=2 40 = 27712) ► DET I 2 = 1414 ( cosit tisin o) 72.=:-1,414 at rE 3 ; YO ILIT → tan Z z = 1.414 (cos 39+isin 37 ) Zz 3-living i Therefore, all complex solutions are To = 1.415 1.414 i 2 = Zz =:-1.414 Log 3 -zoyin i (2) z=3-21 ) Both sides applying 'square root 5 3-2i z = I 13* nors •√3-21 z = + 13-2i To, =