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

For the product, we have
Z_1 × Z_2 = 10 ∠60°× 5 ∠45°∠ = 50 ∠105°
Dividing the numbers, we have

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
Z_1 = 10 ∠60° = 10 \cos(60°) + j10 \sin(60°) = 5 + j8.66
Z2 = 5 \angle 45° = 5 \cos(45°) + j5 \sin(45°) = 3.54 + j3.54

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

|Z_s| = \sqrt{(8.54)^2 + (12.2)^2} = 14.9
\tan \theta _s = \frac{12.2}{8.54}= 1.43

Taking the arctangent of both sides, we have
θ_s = \arctan(1.43) = 55°
Because the real part of Zs 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°