Polar Coordinate System

The position of a point in the plane can be set not only by its Cartesian coordinates `x, y` , but also in other ways. Let′s connect, for example, the point `M` with the origin `O` and consider next two numbers: the length of the segment `OM=r` and angle φ of tilt of this segment to the positive direction of the axis `Ox` (this angle will be positive if the rotation from the axis `Ox` to its shift with the direction `OM` is counterclockwise, and negative otherwise). The segment `r=OM` is called the polar radius of the point `M`, the angle `phi` is its polar angle, the pair of numbers `(r,phi)` are polar coordinates, the point `O` is pole, axis `Ox` is polar axis. Such coordinate system is called polar.

We have the points with the polar coordinates: `A (1;0),B (3/5;-pi/2),C (1/2;(3pi)/4),D (3/5;pi)` .

When we know the polar coordinate of the point we can find its cartesian coordinates by the formulas, that directly follow from the definition of trigonometric functions.

`x=r cos(phi)`, `y=r sin(phi)`

Other way round, if we have cartesian coordinates of the point we can find its polar coordinates by the formulas:

`r=sqrt(x^2+y^2)` ,

`cos(phi)=x/r-x/sqrt(x^2+y^2)` , `sin(phi)=y/r-y/sqrt(x^2+y^2)`

Example. We need to find polar coordinates of point`M` `(-4;4sqrt(3))`.

Using the first of the formulas we find `r=sqrt((-4)^2+(4sqrt(3))^2)=sqrt(16+48)=8`

Then according to the second and third formulas we have `cos(phi)=(-4)/8=1/2` , `sin(phi)=((4sqrt(3))/8)=(sqrt(3))/2` , it follows that `phi=(2pi)/3` . So, `M (8;(2pi)/3)` .