Question

How do you write the trigonometric form of `-4+2i`?

Answer 1

`2sqrt5(cos(2.678)+isin(2.678))`

Explanation:

`"to convert from "color(blue)"cartesian to polar form"`

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

`• r=sqrt(x^2+y^2)`

`• theta=tan^-1(y/x) ; -pi< theta <=pi`

`"here " x=4" and " y=2`

`rArrr=sqrt((-4)^2+2^2)=sqrt20=2sqrt5`

`"note " -4+2i" is in the second quadrant and so " theta" must"`
`"be in the second quadrant"`

`theta=tan^-1(1/2)~~0.464larrcolor(red)"related acute angle"`

`rArrtheta=(pi-0.464)=2.678larrcolor(red)" in second quadrant"`

`rArr-4+2ito2sqrt5(cos(2.678)+isin(2.678))`