Let us write the two complex numbers in polar coordinates and let them be
#z_1=r_1(cosalpha+isinalpha)# and #z_2=r_2(cosbeta+isinbeta)#
Here, if two complex numbers are #a_1+ib_1# and #a_2+ib_2# #r_1=sqrt(a_1^2+b_1^2)#, #r_2=sqrt(a_2^2+b_2^2)# and #alpha=tan^(-1)(b_1/a_1)#, #beta=tan^(-1)(b_2/a_2)#
Their division leads us to
#{r_1/r_2}{(cosalpha+isinalpha)/(cosbeta+isinbeta)}# or
#{r_1/r_2}{(cosalpha+isinalpha)/(cosbeta+isinbeta)xx(cosbeta-isinbeta)/(cosbeta-isinbeta)}#
#(r_1/r_2){(cosalphacosbeta+sinalphasinbeta)+i(sinalphacosbeta-cosalphasinbeta))/((cos^2beta+sin^2beta))# or
#(r_1/r_2)*(cos(alpha-beta)+isin(alpha-beta))# or
#z_1/z_2# is given by #(r_1/r_2, (alpha-beta))#
So for division complex number #z_1# by #z_2# , take new angle as #(alpha-beta)# and modulus the ratio #r_1/r_2# of the modulus of two numbers.
Here #3+4i# can be written as #r_1(cosalpha+isinalpha)# where #r_1=sqrt(3^2+4^2)=sqrt25=5# and #alpha=tan^(-1)(4/3)#
and #1+4i# can be written as #r_2(cosbeta+isinbeta)# where #r_2=sqrt(1^2+4^2)=sqrt17# and #beta=tan^(-1)4#
and #z_1/z_2=5/(sqrt17)(costheta+isintheta)#, where #theta=alpha-beta#
Hence, #tantheta=tan(alpha-beta)=(tanalpha-tanbeta)/(1+tanalphatanbeta)=(4/3-4)/(1+4/3xx4)=(-8/3)/(19/3)=-8/19#.
Hence, #(3+4i)/(1+4i)=5/sqrt17(costheta+isintheta)#, where #theta=tan^(-1)(-8/19)#
