Question

How do you write the complex number `6-8i` in polar form?

Answer 1

`(10,-0.927)`

Explanation:

To express a `color(blue)"complex number in polar form"`

`"that is " (x,y)to(r,theta)`

`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)|)))`
`" where " -pi< theta<=pi`

`"here " x=6" and " y=-8`

`rArrr=sqrt(6^2+(-8)^2)=sqrt(36+64)=10`

Now, 6 - 8i is in the 4th quadrant so we must ensure that `theta"` is in the 4th quadrant.

`theta=tan^-1(-8/6)=-0.927larr" in 4th quadrant"`

`rArr(6,-8)to(10,-0.927)to(10,-53.13^@)`