<prompt> Tafuta idadi ya nambari changamano z zinazotosheleza z 1 nakushoto fraczoverlinez fracoverlinezz kulia 1.  <response>: Tangu z 1, z ei theta kwa pembe fulani theta. Kishaanzaalignkushoto fraczoverlinez fracoverlinezz kulia kushoto fracei thetae-i theta frace-i thetaei theta kulia e2i theta e-2i theta cos 2 theta i sin 2 theta cos 2 theta - i sin 2 theta 2 cos 2 theta.endalignKwa hivyo, cos 2 theta pm frac12.Kwa cos 2 theta frac12, kuna suluhu nne kati ya 0 na 2 pi, yaani fracpi6, frac5 pi 6, frac7 pi6, na frac11 pi6.Kwa cos 2 theta -frac12, kuna suluhu nne kati ya 0 na 2 pi, yaani fracpi3, frac2 pi3, frac4 pi3, na frac5 pi3.Kwa hiyo, kuna 8 ufumbuzi katika z. 