If $z = (4\sqrt{3} - 4 i)^3$, determine $\arg z$. How to find out this $\arg z$? i need help. thanks a lot...
How to figure out the principal argument for z?
3 Answers
We can also use DeMoivre's Theorem here: we have $(4\sqrt{4}-4i)^3=4^3\left(2(\frac{\sqrt{3}}{2}-\frac{1}{2}i)\right)^3=8^3(\cos(-30^\circ)+i\sin(-30^\circ))^3.$ Now, by DeMoivre's Theorem, this reduces to $8^3(\cos(-90^\circ)+i\sin(-90^\circ)),$ so the principal argument is $-90^\circ$ or $-\frac{\pi}{2}.$
Hint: $\arg (z_1z_2)=\arg (z_1)+\arg (z_2)$
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0I think principal argument is usually denoted by 'Arg' – 2012-12-15
Lets use $Arg(z),arg(z)$ as the principal & the general argument $z$ respectively.
$arg(4\sqrt3-4)$ is $n\pi+\arctan\left( \frac{-4}{4\sqrt3}\right)=n\pi-\arctan\frac1{\sqrt3}=n\pi-\frac{\pi}6$ where $n$ is any integer.
$arg\left((4\sqrt3-4)^3\right)$ is $3(n\pi-\frac{\pi}6)=3n\pi-\frac{\pi}2$ as $arg(z\cdot w)=arg(z)+arg(w)$
As the principal argument in $(-\pi,\pi), Arg(4\sqrt3-4)^3$ will be $-\frac{\pi}2$ (putting $n=0$)