Answer
$x-\sqrt{3}y=4$
Work Step by Step
Apply additive identity for cosine
$r\displaystyle \cos(\theta+\frac{\pi}{3})=r(\cos\theta\cos\frac{\pi}{3}-\sin\theta\sin\frac{\pi}{3})$
$\displaystyle \qquad =\frac{1}{2}r\cos\theta-\frac{\sqrt{3}}{2}r\sin\theta$
Apply the conversion formula $(x,y)=(r\cos\theta,r\sin\theta)$
$r\displaystyle \cos(\theta+\frac{\pi}{3})=\frac{1}{2}x-\frac{\sqrt{3}}{2}y$
So the line, in Cartesian coordinates, is
$\displaystyle \frac{1}{2}x-\frac{\sqrt{3}}{2}y=2 \qquad/\times(2)$
$x-\sqrt{3}y=4$
