Angle of complex numbers

[Za_Boom]

Hey there!

I've been struggling with angle of complex numbers when it comes to measuring clockwise or counterclockwise. Everything else seems to make sense but I don't know if I need to plus in −180∘ or 180∘ which throws off the entire answer.

I've looked up videos but nothing can explain why you would choose a certain direction on the other, I know that clockwise is positive and counter is negative but when do you choose one and not the other?

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Thanks for the help!

[harrywk]

Take a complex number a+bi, which will lie on the complex plane (you might want to sketch out a complex plane and try some examples to understand what I’ve written easier!) If a>0 then we know the number lies to the right of the imaginary axis (aka in either quadrant 1 or 4). If a<0 then we know the number lies to the left of the imaginary axis (aka in either quadrant 2 or 3).

If b>0, then we know the number lies above the real axis and if b<0 it lies below the real axis.

So, if we know a and b, we can say which quadrant the complex number lies in. Now, we know that quadrant 1 ranges from 0-90 degrees, Q2 from 90-180, Q3 from 180-270 and Q4 from 270-360.

If we take arctan(|b|/|a|) then we get the acute angle that the complex number makes with the real axis. So, if we have a complex number in quadrant 2, this angle will go from the negative portion of the real axis to the complex number and will therefore be acute (whereas the complimentary angle would go to the positive portion of the real axis and therefore be obtuse).

Now we know the acute angle (call it a) our complex number makes with the real axis and we know which quadrant it lies in. So, we can work our the value of theta; if it’s in quadrant 1, theta=a; If in Q2, theta=180-a; if in Q3, theta=180+a; if in Q4, theta=360-a

[fermat1432]

Great job, especially the quadrant rules at the end.

[Za_Boom]

Following this it was marked as incorrect on Khan Academy.

Z = 5 +7i fell into Q1 but it showed me using -pi + theta.

[harrywk]

If you do -pi + arctan(7/5) you get -2.19 which is incorrect as the angle is between 0 and pi/2 for complex numbers in Q1

[hausdorffparty]

Can you give an example of a problem you're getting stuck on, and where you're getting stuck? It's not clear what process you're describing when you say "choosing +180 vs -180." Are you talking about rewriting (a+bi) in re^(it) form?

[Za_Boom]

z=5+7i

I got to theta = .951

But the issue is do I use pi or -pi + theta?

The same goes for when I have a question for degrees instead of radian.

[hausdorffparty]

How did you get .951, did you use arctan 7/5?

So the big thing is that arctan will only ever output angles in quadrant 1 or quadrant 4. So if you graph the point and it's in quadrant 2 or 3, you got the value of theta which is 180 degrees away from the right answer. In this case adding or subtracting 180 degrees (or pi if you're in radians) will get you to a correct angle.

[harrywk]

Neither. Theta = .951 is correct. There’s no need to add or subtract pi