We start by writing numerator and denominator in polar form
#z=z_1/z_2#
The polar form of a complex number is
#z=r(costheta+isintheta)....................#(1)#
The numerator is
#z_1=3+4i#
#r_1=|z_1|=sqrt((3)^2+(4)^2)=sqrt(9+16)=sqrt25=5#
Therefore,
#z_1=5(3/5+4/5i)#
Comparing this equation to equation #(1)#
#costheta=3/5# and #sintheta=4/5#
So,
we are in the quadrant #I#
#theta=53.13^@#
The polar form is
#z_1=5(cos(53.13^@)+isin(53.13^@))=5e^(53.13i)#
The denominator is
#z_2=1+4i#
#r_2=|z_2|=sqrt((1)^2+(4)^2)=sqrt(1+16)=sqrt17#
Therefore,
#z_2=sqrt17(1/sqrt17+4/sqrt17i)#
Comparing this equation to equation #(1)#
#costheta=1/sqrt17# and #sintheta=4/sqrt17#
So,
we are in the quadrant #I#
#theta=75.96^@#
The polar form is
#z_2=sqrt17(cos(75.96^@)+isin(75.96^@))=sqrt17e^(75.96i)#
Terefore,
#z=z_1/z_2=(5e^(53.13i))/(sqrt17e^(75.96i))#
#=(5/sqrt17)e^((53.13-75.96)i)#
#=(5/sqrt17)e^((-22.83^@)i)#
#=5/sqrt17(cos(-22.83^@)+isin(-22.83^@))#
#=5/sqrt17(0.92-0.39i)#
