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)
