# How do you write the trigonometric form of 5/2(sqrt3-i)?

Jan 20, 2018

The answer is $= 5 \left(\cos \left(- \frac{\pi}{6}\right) + i \sin \left(- \frac{\pi}{6}\right)\right)$

#### Explanation:

The trigonometric form of a complex number $z = a + i b$ is

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

where

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

and

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

Here $z = \frac{5}{2} \left(\sqrt{3} - i\right)$

$| z | = \frac{5}{2} \sqrt{3 + 1} = \frac{5}{2} \cdot 2 = 5$

$z = 5 \left(\frac{\sqrt{3}}{2} - \frac{1}{2} i\right)$

$\cos \theta = \frac{\sqrt{3}}{2}$

$\sin \theta = - \frac{1}{2}$

$\theta = - \frac{\pi}{6}$, $\left[\mod 2 \pi\right]$

Therefore,

The trigonometric form is

$z = 5 \left(\cos \left(- \frac{\pi}{6}\right) + i \sin \left(- \frac{\pi}{6}\right)\right) = 5 {e}^{- i \frac{\pi}{6}}$