For 1) cant help think its a bit long.
* To get back to cartesian just multiply the mod by cos(arg) and sin(arg) to get the real and imaginary componets respectively. Its just polar form.
* To get the cube root, calc the mod arg form, then principle root is mod^(1/3) and arg/3. Then the other two root are same mod and arg+/- 2pi/3 .
2) Id expand the mulitple angles. So expand sin(2x) then factorise the sin part and reason about the cos -part (expand the cos(3x))
3) agree looks llike its harmonic form