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

Aug 4, 2015

$\left(x , y\right) = \left(r \cos \theta , r \sin \theta\right) = \left(6 \cos \left(\frac{2 \pi}{3}\right) , 6 \sin \left(\frac{2 \pi}{3}\right)\right)$

$= \left(6 \cdot - \frac{1}{2} , 6 \cdot \frac{\sqrt{3}}{2}\right) = \left(- 3 , 3 \sqrt{3}\right)$

#### 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:

$r = \sqrt{{x}^{2} + {y}^{2}}$
$\theta = \text{atan2} \left(y , x\right)$

where $\text{atan2} \left(y , x\right)$ 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") :}