MHB Reviewing Phasors: When to Add +/- 180 in $\tan^{-1}$?

[Dethrone]

I'm reviewing phasors (in circuits) and my prof wrote that if you're taking the inverse tangent, $\tan^{-1}{\frac{b}{a}}$ where $a$ is negative, you need to add $+/- 180$. Now I understand that the inverse tangent is defined between $-\pi/2$ to $\pi/2$ for invertibility, etc, but adding or subtracting 180 doesn't always work?

For example, consider $-3+1j$, where (j is the imaginary unit), then the angle is easily seen to be $108.4$ degrees. Using a calculator, $\tan^{-1}(-1/3)=-18.4$ degrees, but no scalar multiply of $180$ will bring me to $108.4$. (I have to add $90$ degrees) So does that imply that my prof's method doesn't generally work?

This is not a problem when you do conversions by hand, but it's been recommended to use the calculator to convert from rectangular to polar, so it would be crucial to know whether or not to add 90, 180, etc.

 

[MarkFL]

You keep editing your post...every time I go to reply, it has changed...:D

For the complex number:

$$z=-1+3j$$

We then find (bearing in mind we are in quadrant II):

$$\arg(z)=\pi-\arctan\left(3\right)\approx108.4^{\circ}$$

You simply mixed up the coordinates in the Argand plane. :D

edit: You changed your post again...:(

 

[Dethrone]

Sorry, LOL, I'm way too tired at the moment (I kept doing $\tan^{-1}\frac{a}{b}$ instead of $\tan^{-1}\frac{b}{a}$which further added to my confusion), but mainly I was trying to find a simple example that matched the question in the book until i realized the book was wrong - so my confusion in the first place was nonsense.

$-1103.55+j353.5$
In polar coordinates:
$1158.79 \angle 162.2$, but the book gets $1158.79\angle 107.76$

The book answer is wrong?

Spoiler

(I normally use the quadrant argument, but when I get rational functions in rectangular form and stick it into a calculator, I can't really distill which quadrant it belongs to, so I decide to stick with adding 180 degrees. I could rationalize it the denominator, but there isn't enough time on the exam :D)

 

[MarkFL]

Just busting your chops, my friend...I have days where nothing I post is right the first time...:D

W|A agrees with you:

View attachment 4216

 

[Nono713]

You basically need atan2 - Wikipedia, the free encyclopedia (scroll down to "definition and computation" to see how it is defined, watch out for the order of the arguments). This will make sure to return the right angle in every case, assuming that zero radians is along the positive real axis and you measure counterclockwise.

 

[Dethrone]

Haha, thanks so much, Mark! :D

EDIT: ...and Bacterius! I never knew about atan2 (pretty cool stuff!), sadly the calculator mandated by my department lacks that function.
EDIT2: Yes! atan2 defined piece-wisely is what I need. (Cool)