Question

Can someone please help me solve this problem!!

Find | z| and Arg( z) for

z=3/(4+2i)
|z|=
Arg(z)=
(Recall −pi< Arg( z)<=pi )

Answer 1

mod(z) or |z| is the distance of the line joining the (0,0) to z when z is drawn in the complex plane. Arg(z) is the angle from the positive x-axis to the ray from (0,0) to z in the complex plane. (Arg(z) is the principle angle, so it must be in the range you specified.)

So z=3/(4+2i) = 3/(4+2i)*(4-2i)/(4-2i) = 3(4-2i)/(20) = 3(2-i)/10 = 3/5 - i3/10
Thus |z|= sqrt[(3/5)^2 + (-3/10)^2]. (You can simplify.)
and Arg(z) =arctan[(-3/10)/(3/5)] = -pi/4

Answer 2

Thank you!!