Question

How do you write the complex number in trigonometric form `-1+sqrt3i`?

Answer 1

`2(cos((2pi)/3)+isin((2pi)/3))`

Explanation:

To convert from `color(blue)"complex to trigonometric form"`

That is `x+yitor(costheta+isintheta)`

`color(orange)"Reminder"`

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

here x = - 1 and y `=sqrt3`

`rArrr=sqrt((-1)^2+(sqrt3)^2)=sqrt4=2`

Now `-1+sqrt3 i` is in the 2nd quadrant, so we must ensure that `theta` is in the 2nd quadrant.

`theta=tan^-1(-sqrt3)=-pi/3" in 4th quadrant"`

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

`rArr-1+sqrt3i=2(cos((2pi)/3)+isin((2pi)/3))`