Question A.5: Complex Arithmetic in Polar Form Given Z1 = 10∠60° and Z2 = ...

For the product, we have

Z_1\times Z_2=10\angle 60^\circ\times 5\angle 45^\circ=50\angle 105^\circ

Dividing the numbers, we have

\frac{Z_1}{Z_2}=\frac{10\angle60^\circ}{5\angle 45^\circ}=2\angle 15^\circ

Before we can add (or subtract) the numbers, we must convert them to rectangular form. Using Equation A.14 to convert the polar numbers to rectangular, we get

A\angle\theta=Ae^{j\theta}=A\cos{(\theta)}+jA\sin{(\theta)}        (A.14)

Z_1=10\angle 60^\circ=10\cos{60^\circ}+j10\sin{(60^\circ)}=5+j8.66

Z_2=5\angle 45^\circ=5\cos{45^\circ}+j5\sin{(45^\circ)}=3.54+j3.54

Now, we can add the numbers. We denote the sum as Z_s:

Z_s=Z_1+Z_2=5+j8.66+3.54+j3.54=8.54+j12.2

Next, we convert the sum to polar form:

\left|Z_s\right|=\sqrt{\left(8.54\right)^2+\left(12.2\right)^2  }=14.9

\tan{\theta_s}=\frac{12.2}{8.54}=1.43

Taking the arctangent of both sides, we have

\theta_s=\arctan{(1.43)}=55^\circ

Because the real part of Z_s is positive, the correct angle is the principal value of the arctangent (i.e., 55° is the correct angle). Thus, we obtain

Z_s=Z_1+Z_2=14.9\angle 55^\circ