Question

How do you use the product and/or quotient theorem of complex numbers in polar form to solve real life situations?

Answer 1

Well, I can think of an application when studying circuits in AC. I am not sure if you know about circuits in Alternate Current but the fact that the current continuously changes can cause difficulties in analyzing the performance of the entire circuit or of some of the elements of it. For this reason is better to use complex numbers to represent the response of the various devices in your circuit to avoid dealing with trig. functions that, as you may know, are quite difficult to manipulate. So, basically, the person that deals with AC "likes" to use complex numbers and their representation in polar form to "simplify" the operations and the maths.
As an example you can have the following circuit where you want to find the current and its phase:

As you can see you can get an equivalent impedance `Z_T` which is a parallel of the upper `Z_1` (R+L+C)and lower `Z_2` (R+C) impedance (correspondent, say, to our `R_(eq)` in a normal DC circuit) but do not worry about the derivation of it, the important thing is that it is in complex polar form!
And then you use it into an AC form of Ohm's Law to find `I_T`!!!!

Hope it helps!