How to Write Down w* in Polar Form for a Given Complex Number?

[andrey21]

Homework Statement

Hi guys I have been given a question, write down w* in polar form where w=2< -(pi/3). I can work out the question when it is in cartesian form just not this way, any help woud be great.

Homework Equations

The Attempt at a Solution

 

[arildno]

What does "<" mean??

 

[andrey21]

Not entirely sure that's just the way it is shown in the question. All i know is when i converted it to cartesian form it became 1 - SQRT3 i

 

[HallsofIvy]

You have left two things ambiguous. As arilno implied "<" is not a standard notation but I am going to assume that you meant the complex number is written in the polar form with modulus r= 3 and angle, or "argument", $\theta= \pi/3$.

The other thing that is ambiguous is the *. I am going to assume that you mean "complex conjugate" which is more commonly written $\overline{w}$.

The connection between "Cartesian representation" and "polar representation" is $z= x+ iy= r (cos(\theta)+ i sin(\theta))$ or, equivalently, $z=x+ iy= r e^{i\theta}$ The complex conjugate is gotten, basically, by changing the sign on "i":
$\overline{z}= x- iy= r (cos(\theta)- i sin(\theta))$ which, because cosine is an even function and sine is an odd function, can be written $\overline{z}= x- iy= r (cos(\theta)- i sin(\theta))$$= cos(-\theta)+ i sin(\theta)$.

Similarly, from $z= x+ iy= r e^{i\theta}$, $\overline{z}= x- iy= r e^{-i\theta}$.

In either case, the complex number given by modulus r and argument $\theta$ has complex conjugate given by modulus r and argument $-\theta$.