# How do you write the trigonometric form of -2i?

Apr 17, 2018

#### Explanation:

$\text{ }$
Given the Complex Number: color(blue)(-2i

The standard form of a complex number is color(red)(z = a+bi

So, we have $z = - 2 i = 0 - 2 i$ with $a = 0 \mathmr{and} b = - 2$

The complex number color(blue)(-2i is marked on a Complex Plane

Observe that color(blue)(-2i lies on the imaginary axis and it takes the same position as $A = \left(0 , - 2\right)$ in Quadrant-3.

This point makes a $\textcolor{red}{{270}^{\circ}}$ from the Real Axis measured in the counter-clockwise direction.

$\textcolor{red}{{270}^{\circ}} = \frac{3 \pi}{2}$ Radians.

Using the formula,

color(red)(z=r(cos theta + i sin theta)

$r$ is the Modulus of z, color(red)(|z|=|a+bi|=sqrt(a^2+b^2

$\theta$ is the argument.

we can represent the complex number $Z = 0 - 2 i$ in trigonometric form as follows:

$r$ is the radius, which is $2$ in this problem.

Hence,

$z = 2 \left[\cos \left(\frac{3 \pi}{2}\right) + i \sin \left(\frac{3 \pi}{2}\right)\right]$

Hope it helps.