Chapter 8 Polar Coordinates and Conics
8.1 Polar Coordinates
In this section, we study polar coordinates and their relation to
Cartesian coordinates.
While a point in the plane has just one pair of Cartesian
coordinates, it has infinitely many pairs of polar coordinates.
To define polar coordinates, we first fix an origin O (called the pole) and an initial ray
from O. Then each point P can be located by assigning to it a polar coordinate pair
(r, ) in which r gives the directed distance from O to P and  gives the directed
angle from the initial ray to ray OP.
Polar Coordinates
Example
Example
Example
Find all the polar coordinates of the point P(2, /6).
Polar Equations and Graphs
Examples
(a) r=1 and r=-1 are equations for the circle of radius 1 centered
at O.
(b)= /6, = 7/6, = -5/6 are equations for the line
Examples
Graph the sets of points whose polar coordinates satisfy the
following coordinates.
a) 1 ≤ r ≤ 2 and 0 ≤  ≤ /2
b) -3 ≤ r ≤ 2 and  = /4
c) r ≤ 0 and  = /4
d) 2/3 ≤  ≤ 5/6
Relating Polar and Cartesian Coordinates
Examples
Examples
Find a polar equation for the circle x2+(y-3)2=9
Examples
Replace the following polar equations by equivalent Cartesian
equations, and identify their graphs.
(a) rcos = - 4
(b) r2 = 4rcos
(c) r = 4/(2cos-sin)
8.2 Graphing in Polar Coordinates
This section describes techniques for graphing equations in polar coordinates. The
Following figure illustrates the standard polar coordinate tests for symmetry.
Symmetry Tests for Polar Graphs
Slope
Note if the graph of r=f() passes through the origin at the value = 0, the
slope of the curve there is tan 0.
Example
Graph the curve r=1-cos .
Example
Graph the curve r2=4cos .
8.3 Areas and Lengths in Polar Coordinates
This section show how to calculate areas of plane regions and lengths of curves
in polar coordinates.
Area of the Fan-Shaped Region
Between the Origin and the Curve
Example
Find the area of the region in the plane enclosed by the cardioid r=2(1+cos ).
Area lying between two polar curves
Area Formula
Example
Find the area of the region that lies inside the circle r=1 and outside the cardioid
r=1-cos .
Length of a Polar Curve
Examples
Find the length of the cardioid r=1-cos .