Q.Find the Cartesian co-ordinates of the point whose polar co-ordinates are $\left(2, \dfrac{\pi}{4}\right)$

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 definitions of sine and cosine in the right triangle formed by dropping a perpendicular from $P$ to the axis. Converting the other way — Cartesian to polar — uses $r=\sqrt{x^2+y^2}$ together with $\cos\theta=x/r$ and $\sin\theta=y/r$ solved simultaneously (not just one of the two) so that $\theta$ lands in the correct quadrant.