# How do you write the complex number in trigonometric form -8-5sqrt3i?

Dec 14, 2017

$z = \sqrt{139} \left(\cos 47.3 + i \sin 47.3\right)$

#### Explanation:

If $z = a + b i$ then $z = r \left(\cos \theta + i \sin \theta\right)$

$r = \sqrt{{a}^{2} + {b}^{2}}$

$\theta = {\tan}^{- 1} \left(\frac{b}{a}\right)$

$a = - 8$
$b = - 5 \sqrt{3}$

$r = \sqrt{{\left(- 8\right)}^{2} + {\left(- 5 \sqrt{3}\right)}^{2}} = \sqrt{139}$

$\theta = {\tan}^{- 1} \left(\frac{- 5 \sqrt{3}}{- 8}\right) \approx = {47.3}^{\circ}$

$z = \sqrt{139} \left(\cos 47.3 + i \sin 47.3\right)$