Question

Express the complex number in trigonometric form.
-6 + 6 sqrt3 i

Answer 1

a+ib=r (cos2pi/3+isin2pi/3)

Explanation:

a+ib=r(cos theta+isin theta)

r=sqrt a^2+b^2

r=sqrt (-6)^2+(6sqrt3)^2

r=12

theta=tan^-1 (y/x)

theta=tan^-1(6sqrt3/ -6)

theta=tan^-1(-sqrt 3)

theta=-60 degrees

Now, we no that theta is in the 2nd quadrant because sin is positive Therfore, we subtract 60 from 180.

180-60=120

theta=120 degrees

Now we can convert 120 degrees to radians: 120 times pi/180=2pi/3

theta=2pi/3 r=12

Substitute: a+ib=r (cos2pi/3+isin2pi/3)

Answer 2

The trigonometric form of the complex number is 12(cos 120° + i sin 120°)

Explanation:

* Lets revise the complex number in Cartesian form and polar form

- The complex number in the Cartesian form is a + bi

-The complex number in the polar form is r(cosФ + i sinФ)

* Lets revise how we can find one from the other

- r² = a² + b²

- tanФ = b/a

* Now lets solve the problem

∵ z = -6 + i 6√3

∴ a = -6 and b = 6√3

∵ r² = a² + b²

∴ r² = (-6)² + (6√3)² = 36 + 108 = 144

∴ r = √144 = 12

∵ tan Ф° = b/a

∴ tan Ф = 6√3/-6 = -√3

∵ The x-coordinate of the point is negative

∵ The y-coordinate of the point is positive

∴ The point lies on the 2nd quadrant

* The measure of the angle in the 2nd quadrant is 180 - α, where

α is an acute angle

∵ tan α = √3

∴ α = tan^-1 √3 = 60°

∴ Ф = 180° - 60° = 120°

∴ z = 12(cos 120° + i sin 120°)

* The trigonometric form of the complex number is

12(cos 120° + i sin 120°)