How do you write the complex number in trigonometric form -2(1+sqrt3i)?

Feb 23, 2017

The trigonometric form is $= - 4 \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right)$

Explanation:

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

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

Here, we have

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

The modulus is

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

$z = - 4 \left(\frac{1}{2} + \frac{\sqrt{3}}{2} i\right)$

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

$\theta = \frac{\pi}{3}$

Therefore,

$z = - 4 \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right)$

$= - 4 {e}^{\frac{\pi}{3} i}$