<prompt> Nambari changamano z rei theta inatoshelezakushoto 2z frac1z kulia 1.Tafuta thamani ya chini zaidi ya sin2 theta.  <response>: Kuweka z rei theta katika equation iliyotolewa, tunapatakushoto 2rei theta frac1r e-i theta kulia 1.Kishakushoto 2r cos theta 2ri sin theta frac1r cos theta - fracir sin theta kulia 1.Hivyo,kushoto 2r cos theta frac1r cos theta kulia2 kushoto 2r sin theta - frac1r sin theta kulia2 1.Kupanua, tunapata4r2 cos2 theta 4 cos2 theta frac1r2 cos2 theta 4r2 sin2 theta - 4 dhambi 2 theta frac1r2 sin2 theta 1,inayorahisisha4r2 4 cos2 theta - 4 sin2 theta frac1r2 1.Kwa kuwa cos2 theta 1 - sin 2 theta,4r2 4 - 4 sin2 theta - 4 sin2 theta frac1r2 1,hivyo8 sin2 theta 4r2 frac1r2 3.Na AM-GM, 4r2 frac1r2 ge 2 sqrt4r2 cdot frac1r2 4, hivyo 8 sin2 ge 7, ausin2 theta ge frac78.Usawa hutokea wakati r frac1sqrt2, hivyo thamani ya chini zaidi ya sin 2 theta ni frac78. 