15 The Principal Argument when Multiplying Complex Numbers

When multiplying two complex numbers in polar form using the principal value of their arguments, is it always the case that adding these two arguments will give the principal value of the argument of the product?
This video illustrates the answer to this question using an animation created with GeoGebra software.
Viewers are invited to solve this question:
Find polar forms for these complex numbers using the principal value of the argument: (a) z = -1 - i (b) w = 1 - i (c) zw
This is followed by the full worked solution.
Previous videos in this series are:
01 What is a Complex Number?
02 Adding, Subtracting and Multiplying Complex Numbers
03 Dividing Complex Numbers
04 Complex Conjugates
05 The Field of Complex Numbers
06 The Complex Plane
07 The Modulus of a Complex Number
08 Distance on the Complex Plane
09 Properties of the Modulus of a Complex Number
10 Complex Numbers and the Unit Circle
11 The Polar Form of a Complex Number
12 The Principal Argument of a Complex Number
13 The Geometrical Effect of Multiplying by a Complex Number
14 Multiplication of Complex Numbers in Polar Form