Question

How do you convert the polar coordinate (6, 2pi/3) into cartesian coordinates?

Answer 1

`(x, y) = (r cos theta, r sin theta) = (6cos((2pi)/3), 6sin((2pi)/3))`

`= (6*-1/2, 6*sqrt(3)/2) = (-3,3sqrt(3))`

Explanation:

Given radius `r` and angle `theta`, the cartesian coordinates `x` and `y` are given by the formulae:

`x = r cos theta`
`y = r sin theta`

Conversely, given `x` and `y`, the radius `r` and angle `theta` are determined by the formulae:

`r = sqrt(x^2+y^2)`
`theta = "atan2"(y, x)`

where `"atan2"(y, x)` is defined as follows:

`"atan2"(y, x) = { (arctan(y/x), " if x > 0"), (arctan(y/x) + pi, " if x < 0 and y >= 0"), (arctan(y/x) - pi, " if x < 0 and y < 0"), (pi/2, " if x = 0 and y > 0"), (-pi/2, " if x = 0 and y < 0"), ("undefined", " if x = 0 and y = 0") :}`