> [!NOTE] Definition (Polar Coordinates) > Let $(x,y)\in \mathbb{R}^{2}$ be an element of the [[Real n-Space|real 2-space]]. Then its polar coordinates are given by $(r,\theta)$ where $x=r\cos\theta$and$y=r\sin\theta$where $r$ is known as the **radial coordinate** and $\theta$ the **angular coordinate.** # Properties As shown in [[Radial Coordinate of Element of Real 2-Space equals Length]], $||(x,y)||=r$ since $\cos ^{2}\theta+\sin^{2}\theta=1.$ On the other hand [[Angular Coordinate of Element of Real 2-Space equals Angle measured from x-axis]] asserts that $\theta = \angle \underline{i} \underline{v}$ where $\underline{v}=(x,y)$ and $i=(1,0).$