Q.Find the Cartesian co-ordinates of the point whose polar co-ordinates are $\left(\dfrac{1}{2}, \dfrac{7\pi}{3}\right)$

Polar co-ordinates are a second, non-Cartesian way to fix the position of a point in a plane. A fixed point $O$ (the pole) and a fixed ray $OX$ from it (the polar axis) are chosen; any other point $P$ is then located by the ordered pair $(r,\theta)$, where $r=OP$ is the radius vector (a non-negative length) and $\theta$ is the vectorial angle that $OP$ makes with $OX$, taken as the smallest non-negative angle so that $0\le\theta<2\pi$. The pole itself is the one point in the plane with no polar co-ordinates, since it has no well-defined direction. Overlaying a Cartesian frame with its origin at the pole and its X-axis along the polar axis gives the conversion formulas $x=r\cos\theta$ and $y=r\sin\theta$, derived directly from the definiti …