Polar Coordinates
So far, you have been representing graphs as collections of points on the
rectangular coordinate system. The corresponding equations for these graphs have been
in either rectangular or parametric form. In this section, you will study a coordinate
system called the polar coordinate system.
(a) (b) (c)
With rectangular coordinates, each point has a unique representation. This is
not true with polar coordinates. For instance, the coordinates
and
and
represent the same point. In general, the point can be written as
or
where n is any integer. Moreover, the pole is represented by 0, , where is any angle.
r, r, 2n 1
r, r, 2n
r,
r, r,
r
r, r, 2
x, y
π
| 2 3 = 11 θ |
0
π2
3
6
11
π2
π
6
π
3, ) )
π
| 2 3 = θ π |
0
π2
3
-
6
π2
π
3, ) -6)
π 0
π2
3
=
3
| 2, ) |
| 1 2 3 |
)
π2
π
π3
θ
directed angle, counterclockwise from polar axis to segment OP
r directed distance from O to P
P r, ,
O
O,
x, y
O
= directed angle
Polar
axis
P = (r, )
r = directed distance
θ
θ
To form the polar coordinate system in the plane, fix a point called the pole (or
origin), and construct from an initial ray called the polar axis, as shown in
Figure 2.1. Then each point in the plane can be assigned polar coordinates
as follows
