Question

Express the complex number in trigonometric form.

-6 + 6\sqrt(3) i

Answer 1

12(cos120°+isin120°)

Explanation:

The rectangular form of a complex number is expressed as z = x+iy

where the modulus |r| =
`\sqrt{x^(2)+y^(2)` and the argument
`\theta = tan^(-1)(y)/(x)`

In polar form, x =
`rcos\theta \ and\ y = rsin\theta`

`z = rcos\theta+i(rsin\theta)\\z = r(cos\theta+isin\theta)`

Given the complex number,
`z = -6+6√(3) i`. To express in trigonometric form, we need to get the modulus and argument of the complex number.

`r = \sqrt{(-6)^(2)+(6√(3) )^(2)}\\r = √(36+(36*3)) \\r = √(144)\\ r = 12`

For the argument;

`\theta = tan^(-1) (6√(3) )/(-6) \\\theta = tan^(-1)-√(3) \\\theta = -60^(0)`

Since tan is negative in the 2nd and 4th quadrant, in the 2nd quadrant,

`\theta =180-60\\\theta = 120^(0)`

z = 12(cos120°+isin120°)

This gives the required expression.