We start by writing numerator and denominator in polar form
z=z_1/z_2z=z1z2
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)
