Polar Coordinates

Although we introduced  polar coordinates in the previous section, this coordinate transformation occurs frequently enough to warrant our considering it separately.  Let's begin by reviewing what it means to assign polar coordinates to a point the in the plane.

Suppose that l is a ray that begins at an origin O, and suppose that P is a point in the plane. Then P can be located with respect to l and O by specifying both the distance r from O to P and an angle q  formed by the segment OP and the ray l.

The order pair ( r, q)  is called the polar coordinates of the point P.  

The polar coordinates of a point are not unique.  Since angles repeat every 2p radians, it follows that ( r,q) = ( r,q+2p) . Moreover, a negative value of r implies a rotation of 180^°, so that we also have ( -r,q) = ( r,q+p) .       

EXAMPLE 1    Locate the points ( 2,p/4) and ( -3,p/3) using polar coordinates.       

Solution: The point with polar coordinates ( 2,p/4) must be a distance of 2 from the origin along the ray which is at an angle of p/4 from the x-axis. Likewise, the point with polar coordinates ( -3,p/3) is the same as the point ( 3,p+p/3) , as is shown below:

       

If we choose the ray l to be the positive x-axis, then a point  P in the plane has both cartesian coordinates ( x,y) and polar coordinates ( r,q) .

The definition of the sine and cosine functions imply that (x,y) is given in terms of ( r,q) by

EXAMPLE 2    Convert the point ( 4,p/4) from polar coordinates into cartesian coordinates.       

Solution: To do so, we let r = 4 and let q = p/4 in (1) to obtain

x = 4cos æ è  p 4 ö ø = 2Ö2,        y = 4sin æ è  p 4 ö ø = 2Ö2

       

EXAMPLE 3    Convert the point ( Ö3,-1) from cartesian coordinates into polar coordinates.       

Solution: To do so, we let x = Ö3 and let y = -2 in (2) to obtain

r2 = 3+1 = 4,    r = 2 tan( q) = -  1 Ö3 ,        q = -p/6