Answer
$z=10(\displaystyle \cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3})$
Work Step by Step
See p. 604,
A complex number $z=a+bi$ has the polar (or trigonometric) form
$z=r(\cos\theta+i\sin\theta)$
where $r=|z|=\sqrt{a^{2}+b^{2}}$ and $\tan\theta=b/a$.
The number $r$ is the modulus of $z$, and $\theta$ is an argument of $z$.
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$z=-5+5\sqrt{3}i$
$r=|z|=\sqrt{(-5)^{2}+(5\sqrt{3})^{2}}=\sqrt{25+25\cdot 3}=10$
$\displaystyle \tan\theta=\frac{5\sqrt{3}}{-5}=-\sqrt{3}$,
In quadrant I, $\displaystyle \tan\frac{\pi}{3}=\sqrt{3}$.
By symmetry (on the unit circle),
$\theta$ can be $\displaystyle \frac{2\pi}{3}$ (quadrant II) or $\displaystyle \frac{5\pi}{3}$ (quadrant IV)
$z=-5+5\sqrt{3}i$
lies in quadrant II, so we select the appropriate argument:
$z=10(\displaystyle \cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3})$
