Question

How do you convert the Cartesian coordinates `(-1/2, -sqrt(3)/2)` to polar coordinates?

Answer 1

Polar coordinates of `(-1/2,-sqrt3/2)` are `(1,(4pi)/3)`.

Explanation:

The relation between Cartesian coordinates `(x,y)` and polar coordinates `(r,theta)` is `x=rcostheta` and `y=rsintheta`.

As such `r=sqrt(x^2+y^2)` and hence for `(-1/2,-sqrt3/2)`,

`r=-sqrt((-1/2)^2+(-sqrt3/2)^2)=-sqrt(1/4+3/4)=sqrt1=1`

Hence `costheta=x/r=-1/2` and `sintheta=y/r=-sqrt3/2`.

Hence `theta` is in third quadrant and `theta=(4pi)/3`.

Therefore polar coordinates of `(-1/2,-sqrt3/2)` are `(1,(4pi)/3)`.