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Complex Argument
AHL 1.12
The argument of a complex number is the angle that it forms with the real (x) axis on the complex plane:
By noticing a right angled triangle, we can say that
tan(argz)=ab🚫
When a>0:
argz=arctan(ab)🚫
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question, and often both will be accepted.
Complex argument when a<0
AHL 1.12
If a<0, then argz is in the second or third quadrant, which are not in the range of arctan. We therefore need to add or subtract π to get the correct argument:
When a<0:
arg(z)=arctan(ab)±π🚫
When z is in the second quadrant, we add π; when z is in the third quadrant, we subtract π.
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question, and often both will be accepted.
Complex Argument when a=0 (purely imaginary)
AHL 1.12
If a=0, then tan(argz)=ab is undefined. tanθ is also undefined for θ=2π,23π… So when a=0 we have
arg(bi)=⎩⎪⎪⎪⎨⎪⎪⎪⎧2πb>0 −2πb<0🚫
This can be seen on the complex diagram by remembering that bi lies on the yi axis:
Note on convention: By convention, the argument is usually given in radians in the range [−π,π]. It will be made clear by the IB which range is preferred in a given question.
Nice work completing Complex Argument, here's a quick recap of what we covered:
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