Question

How do you write the complex number in trigonometric form `4-4sqrt3i`?

Answer 1

`z=8[cos((2pi)/3)+isin((2pi)/3)]`

Explanation:

`(x,y)tor(costheta+isintheta)`

`color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(r=sqrt(x^2+y^2))color(white)(2/2)|)))`

`"and " color(red)(bar(ul(|color(white)(2/2)color(black)(theta=tan^-1(y/x))color(white)(2/2)|)))`

`"here "x=4" and " y=-4sqrt3`

`rArrr=sqrt((4)^2+(-4sqrt3)^2)=sqrt64=8`

`"now " 4-4sqrt3` is in the 2nd quadrant so we must ensure that
`theta` is in the 2nd quadrant.

`rArrtan^-1((4sqrt3)/4)=tan^-1(sqrt3)=pi/3`

`rArrtheta=(pi-pi/3)=(2pi)/3larrcolor(red)" in 2nd quadrant"`

`rArr4-4sqrt3i=8[cos((2pi)/3)+isin((2pi)/3)]`