Find z^n+ 1/z^n: Why Consider Only One Argument?

[Magnetons]

Homework Statement

compute $z^n+ \frac 1 z^n$
if $z+\frac 1 z= \sqrt3$

Relevant Equations

No Equation

Firstly I converted the given equation to a quadratic equation which is
$z^2- (\sqrt3)z+1=0$
I got two solutions:
1st sol $z=\frac {(\sqrt3 + i)} {2}$
2nd sol $z=\frac {(\sqrt3 -1)} {2}$
Then I found modulus and argument for both solution . Modulus=1 Arguments are $\frac {\Pi} {6}$ and $\frac {11* \Pi } {6}$

My question is {Why we consider only one argument $\frac {\Pi} {6}$ and not other}

 

[PeroK]

My question is \mathbf {Why we consider only one argument \frac {\Pi} {6} and not other}

First, let $z_0$ be a solution to $z + \frac 1 z = w$. Then, $\frac 1 {z_0}$ is also a solution. The two solutions are, therefore, essentially the same. You can check, if you want, that your two solutions are inverses of each other.

 

[Magnetons]

First, let $z_0$ be a solution to $z + \frac 1 z = w$. Then, $\frac 1 {z_0}$ is also a solution. The two solutions are, therefore, essentially the same. You can check, if you want, that your two solutions are inverses of each other.

Got it

 

[PeroK]

And, of course, if you want to compute $f(z) = z^n + \dfrac 1 {z^n}$, then $f(z_0) = f(\frac 1 {z_0})$. And you get the same value for both solutions.