Week 8
Outline
Polar Coordinate System
Examples of Polar Curves
Tangents to Polar Curves
Intersections of Polar Curves
Areas Bounded by Polar Curves
Arc Lengths for Polar Curves
Polar Coordinate System
To describe a point in the plane in polar coordinates, we give its distance from the
origin, called
r
, and its angle in radians going counterclockwise from the positive
x
axis,
which we call
θ
. So, instead of using (
x, y
) in rectangular coordinates, we describe a point
by (
r, θ
).
Remarks
1. In polar coordinates, a point can have more than one description. The point (1
,
0) in
rectangular coordinates has distance 1 from the origin and has angle 0 with the positive
x
axis. So in polar coordinates, it is (
r, θ
) = (1
,
0). But it is also given by (1
,
2
π
), (1
,
4
π
),
etc. Even wackier is the origin, which is given by (0
,
0
.
9
π
), (0
,
2
.
3
π
), (0
,
π
170
), etc. That is
to say, (
r, θ
) can also be represented by (
r, θ
+ 2
nπ
) and the origin by (0
, θ
) for any
θ
.
2.
Sometimes we may extend the meaning of polar coordinates (
r, θ
) to the case in
which
r
is negative by agreeing that, the points (

r, θ
) and (
r, θ
) lie on the same line
through the origin and at the same distance

r

from the origin, but on opposite sides of
the origin. In short, (

r, θ
) represents the same point as (
r, θ
+
π
).
Relationship between Rectangular and Polar Coordinates
By Pythagorean theorem, we have the following basic equations
x
=
r
cos
θ, y
=
r
sin
θ, r
=
p
x
2
+
y
2
.
Also,
tan
θ
=
y
x
,
assuming
x
6
= 0
.
Remark
1
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These four equations can be easily read off the picture and so there is no need to
memorize them.
Examples
1. Plot the points whose polar coordinates are given.
(a) (1
,
5
π
4
), (b) (2
,
3
π
), (c) (2
,

2
π
3
)
2. Represent the given point with rectangular coordinates in terms of polar coordinates.
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 Fall '06
 GROSS
 Calculus, Arc Length, Cartesian Coordinate System, Polar Coordinates, Coordinate system, Polar coordinate system, Conic section, dr dθ dr dθ

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