Answer:

i)
ii) 
Thinking Process:
This can either be solved using complex variables or simply trigonometry (they are all the same in a way)

But we'll simply go with trigonometry and good old Pythagorean theorem.
Solution:
For Cartesian coordinates are: 
To covert them into 
We can use:
- The Pythagoras theorem to find


- Arctangent to find


These are related since:


So, let's start solving:
Part a) (4,-4)
For r:




i) for
we'll go with
ii) for
we'll go with
For theta:




One thing to keep in mind is that the point (4,-4) lies in the 4th quadrant of the xy-plane, and the range given for
is
. So we have to add
to our answer (we have gone around the circle and stopped at the same place as
radians)
radians
radians
The Answer for part(a) is 
Part b) (-1, sqrt(3))
For r:



i)
for 
ii)
for
For theta:


this not in range of theta:
.
and the point
lies in the 2nd quadrant, so we can add
radians to our answer
in range: 

The Answer for part(b) is 