Question

How do you change the rectangular coordinate `(-6, 6sqrt3)` into polar coordinates?

Answer 1

`(-6,6sqrt3)to(12,(2pi)/3)`

Explanation:

To convert from `color(blue)"rectangular to polar coordinates"`

That is `(x,y)to(r,theta)`

`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 = - 6 and y `=6sqrt3`

`rArrr=sqrt(6^2+(6sqrt3)^2)=sqrt(36+108)=sqrt144=12`

Now `(-6,6sqrt3)` is in the 2nd quadrant so we must ensure that `theta` is in the 2nd quadrant.

`theta=tan^-1((6sqrt3)/-6)=tan^-1(-sqrt3)`

`=-pi/3" in 4th quadrant"`

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

`rArr(-6,6sqrt3)to(12,(2pi)/3)`