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 multiplication leads us to
#{r_1r_2}{(cosalpha+isinalpha)(cosbeta+isinbeta)}# or
#(r_1r_2){(cosalphacosbeta+i^2sinalphasinbeta)+i(sinalphacosbeta+cosalphasinbeta))# or
#(r_1r_2){(cosalphacosbeta-sinalphasinbeta)+i(sinalphacosbeta+cosalphasinbeta))# or
#(r_1r_2)*(cos(alpha+beta)+isin(alpha+beta))# or
#z_1*z_2# is given by #(r_1r_2, (alpha+beta))#
So for multiplication complex number #z_1# by #z_2# , take new angle as #(alpha+beta)# and modulus is #r_1*r_2# i.e. product of the modulus of two numbers.
Here #4+3i# can be written as #r_1(cosalpha+isinalpha)# where #r_1=sqrt(4^2+3^2)=sqrt25=5# and #alpha=tan^(-1)3/4#
and #-3+2i# can be written as #r_2(cosbeta+isinbeta)# where #r_2=sqrt((-3)^2+2^2)=sqrt13# and #beta=tan^(-1)(-2/3)#
and #z_1*z_2=5sqrt13(costheta+isintheta)#, where #theta=alpha+beta#
Hence, #tantheta=tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)=(3/4+(-2/3))/(1-3/4xx(-2/3))=(1/12)/(3/2)=1/18#.
Hence, #(4+3i)*(-3+2i)=5sqrt13(costheta+isintheta)#, where #theta=tan^(-1)(1/18)#
