# 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 pointM (-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) .