# How do you write the complex number in trigonometric form -8+3i?

Aug 14, 2017

The trigonometric form is $= 2.92 \left(\cos \left({159.4}^{\circ}\right) + i \sin \left({159.4}^{\circ}\right)\right) = 2.92 {e}^{{159.4}^{\circ} i}$

#### Explanation:

Our complex number is

$z = - 8 + 3 i$

The trigonometric form is

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

If our complex number is $z = a + i b$

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

And

$\cos \theta = \frac{a}{|} z |$ and

$\sin \theta = \frac{b}{|} z |$

Therefore,

$| z | = \sqrt{{\left(- 8\right)}^{2} + {3}^{2}} = \sqrt{64 + 9} = \sqrt{73} = 2.92$

$\cos \theta = - \frac{8}{\sqrt{73}}$

$\sin \theta = \frac{3}{\sqrt{73}}$

We are in the Quadrant $I I$

$\Theta = {159.4}^{\circ}$

The trigonometric form is

$z = 2.92 \left(\cos \left({159.4}^{\circ}\right) + i \sin \left({159.4}^{\circ}\right)\right) = 2.92 {e}^{{159.4}^{\circ} i}$