Question

How do you write the trigonometric form of `-1+sqrt3i`?

Answer 1

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

Explanation:

`"to convert from "color(blue)"cartesian to trig. form"`

`"that is " (x,y)tor(costheta+isintheta)`

`• r=sqrt(x^2+y^2)`

`• theta=tan^-1(y/x)`

`"here " x=-1" and " y=sqrt3`

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

`-1+isqrt3" is in the second quadrant so we must ensure "`
`"that " theta" is in the second quadrant"`

`theta=tan^-1(sqrt3)=pi/3larrcolor(red)" related acute angle"`

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

`rArr-1+isqrt3=2(cos((2pi)/3)+isin((2pi)/3))`