Let the quotient be q=3+4iq=3+4i
and the divisor be d=9-4id=9−4i
In trigonometric form:
color(white)("XXX")q=r_q(cos(theta_q)+isin(theta_q))XXXq=rq(cos(θq)+isin(θq))
with
color(white)("XXX")r_q=sqrt(3^2+4^2)=5XXXrq=√32+42=5
and
color(white)("XXX")theta_q= arctan(4/3)~~0.927295XXXθq=arctan(43)≈0.927295
Similaryly, in trigonometric form:
color(white)("XXX")d=r_d(cos(theta_d)+isin(theta_d))XXXd=rd(cos(θd)+isin(θd))
with
color(white)("XXX")r_d=sqrt(9^2+(-4)^2) =sqrt(97)~~9.848858XXXrd=√92+(−4)2=√97≈9.848858
and
color(white)("XXX")theta_d=arctan(9/(-4))~~-0.41822XXXθd=arctan(9−4)≈−0.41822
Using trigonometric form:
color(white)("XXX")q/d = (r_q)/(r_d)(cos(theta_q-theta_d)+isin(theta_q-theta_d))XXXqd=rqrd(cos(θq−θd)+isin(θq−θd))
color(white)("XXX")=5/sqrt(97)(cos(0.927295-(-041822))+isin(0.927295-(-0.41822))XXX=5√97(cos(0.927295−(−041822))+isin(0.927295−(−0.41822))
color(white)("XXX")~~0.51(cos(1.35)+isin(1.35))XXX≈0.51(cos(1.35)+isin(1.35))
