Real plane/Polar coordinates/Angle naive/Example

An angle ${}\alpha$ and a positive real number ${}r$ define a unique point

${}P=(x,y)=(r\cos \alpha ,r\sin \alpha )=r(\cos \alpha ,\sin \alpha )\,$

in the real plane ${}\mathbb {R} ^{2}$. Here, ${}r$ is the distance between the point ${}P$ and the zero point ${}(0,0)$ and ${}(\cos \alpha ,\sin \alpha )$ means the intersecting point of the ray through ${}P$ with the unit circle. Every point ${}P=(x,y)\neq 0$ has a unique representation with ${}r={\sqrt {x^{2}+y^{2}}}$ and with an angle ${}\alpha$, which has to be chosen accordingly (the zero point is represented by ${}r=0$ and an arbitrary angle). The components ${}(r,\alpha )$ are called the polar coordinates of ${}P$.