Expert: Josh - 3/25/2009

Question
Hi - if I have something like -1-i1 and convert it to polar, the angle tan-1(-1/-1) comes out to be 45 degrees.  In actuality it is really in the third quadrant (225 degrees).  Am I doing something wrong?  Is there any way, short of looking at the rectangular form to know which quadrant is the correct one?

Thanks,
Steve


Answer
Hi Steve,

You are right. The complex number "-1-i" lives in the third quadrant 180<x<=270 in the Argand plane. Scientific programming languages usually implement two forms of the inverse tangent function. The first version maps the ratio (-1/-1) to an angle in the interval from -90 to 90 degrees. The second version takes two input arguments, viz., the real and imaginary parts, to give you a unique answer in the right quadrant. If your calculator only allows you to do the former, you will need to correct the ambiguity yourself. There is nothing wrong with what you do. Since tan(x)=tan(x+180) for any angle x, the answer might be out by 180 degrees.

It helps knowing the acronym ASTC. It tells us that
Sin(x) has the same sign in the 1st and 2nd quadrant;
Cos(x) has the same sign in the 1st and 3rd quadrant;
Tan(x) has the same sign in the 1st and 4th quadrant.

Here is an alternate approach:

In your question, the calculator returns the answer tan(-1/-1)=45.
You know the answer might be out by 180 (because we are dealing with the tangent function). So, the possible solutions 45 and 215 are in the 1st and 3rd quadrant.

Recall that a complex number Z=x+iy may be written as Z=rcos(w)+irsin(w), where the modulus r=sqrt(x^2+y^2) and principal argument w=tan^-1(x,y) is the angle in the range from 0 to 360.

We see that the real part is negative (actually x=-1, but the magnitude doesn't matter). Without drawing a diagram, you only have to check whether cos(45) is positive/negative to know which is the right solution. Here, cos(45) is positive. This is not what we want, as the real part of "-1-i" is negative (just the opposite). So, we immediately cease upon w=45+180=215 as the right solution.

It takes a lot of words to explain this, but this method only requires you to check EITHER the real part, or the imaginary part, with respect to one of the angle w. You don't actually need to look at BOTH real and imaginary parts.

e.g., Consider Z = -sqrt(3)+i. Calculator tells you the angle is -30 degree. Observe the real part (-1.57...) is negative. Check: Is cos(-30) negative? Answer: No. Then, -30+180=150 deg. must be the true angle.