Complex number consists of 2 parameters.
They are the modulus (length) and argument (direction).
Argand diagram is the pictoral form of representing this complex number.
In the Argand diagram, quadrants define the position of the "complex" line.
Z = a + ib (click for information) is the general form of writing the complex number.
"a" and "b" will define its polar counterparts, modulus and argument.
Having 4 quadrants in the Argand diagram means having 4 combinations of "a" with "b".
1) Z = a + ib
2) Z = -a + ib
3) Z = -a - ib
4) Z = a - ib
The first case lies in the first quadrant (Q1).
The second case lies in Q2.
Third case lies in Q3, and
fourth case in Q4.
Therefore knowing the sign of "a" and "b" let you know which quadrant the complex line lies.
And this is where the mistake lies!
The argument is always computed wrongly for the Q2, Q3 and Q4.
Only the positive sign of "a"and "b" is taken to get the value of the angle (argument).
Example of error:
Z = 5 + i5 ==> Argument = +450 (Q1)
However, Z = -5 + i5 ==> Also taken as + 450 forcing it to lie in Q1 (wrong!).
It should lie in Q2 since now the real term is negative.
Values of "a" and "b" are not the only parameters needed to find the angle.
Their signs are equally important!
Think of the Argand diagram representation before the definition of the angle.
This will ensure that the angle is correctly calculate later on since you have an idea which quadrant the complex line should lies then.
Doing the complex number and its conversion from rectangular form to polar form properly will make you happy and like maths. Proper thinking process will path you into a good habit that leads to confidence in maths.
Maths is Interesting! And fun ...