I like the idea of using polar notation as taught in the AAC e-books - the relative simplicity of using these complex numbers as you would with basic DC theory.

But does it have its limitations?

For example:
A capacitor C is connected in series with 40Ω resistor across a supply of freq. 60Hz. A current of 3A flows and the circuit Z is 50Ω.
Value of capacitance?

I cant see a way of using polar notation - the only way I can see to solve this question is to have to draw phasor diagrams, then use trig./pythag. theorems?

 

Polar notation is just another way to express complex numbers. That doesn't mean it is always more convenient than the classic representation of x+yj.
Polar notation is more convenient for multiplication and division, were the amplitude is multiplied/divided and the phase is added/substracted.
However, in addition and substraction the cartesian representation is more suitable as you can add/substract the real and imaginary part separately. This is the case in this exercise too.