# How do you convert 5 sqrt 3 - 5i to polar form?

Dec 30, 2016

The answer is $= 10 \left(\cos \left(- \frac{\pi}{6}\right) + i \sin \left(- \frac{\pi}{6}\right)\right)$

#### Explanation:

Let $z = 5 \sqrt{3} - 5 i$

We must transform this equation to the form

$z = r \left(\cos \theta + i \sin \theta\right)$

We calculate the modulus of $z$

∣z∣=sqrt(25*3+25)=sqrt100=10

z=∣z∣((5sqrt3)/(∣z∣)-i5/(∣z∣))

$z = 10 \left(5 \frac{\sqrt{3}}{10} - i \frac{5}{10}\right)$

$z = 10 \left(\frac{\sqrt{3}}{2} - \frac{1}{2} i\right)$

Therefore,

$\cos \theta = \frac{\sqrt{3}}{2}$ and $\sin \theta = - \frac{1}{2}$

We are in the 4th quadrant

$\theta = - \frac{\pi}{6}$

Therefore,

$z = 10 \left(\cos \left(- \frac{\pi}{6}\right) + i \sin \left(- \frac{\pi}{6}\right)\right)$

$z = {e}^{\frac{- i \pi}{6}}$