Jim Lambers
Math 1B
Fall Quarter 2004-05
Leture 23 Notes
These notes orrespond to Setion 7.5 in the text.
Polar Co ordinates and Graphs
Throughout this ourse, we have denoted a point in the plane by an ordered pair (
x; y
), where the
num
bers
x
and
y
denote the direted (i.e., signed positive or negative) distane between the point
and eah of two perpendiular lines, the
x
-axis and the
y
-axis. The elements of this ordered pair
are alled
oordinates
, and the o ordinates used in this partiular method of identifying points in
the plane are alled
Cartesian oordinates
.
In this leture, we introdue an alternative oordinate system known as the
polar oordinate
system
. In this system, a p oint in the plane is identied by an ordered pair (
r;
), where:
r
is the direted distane from a point designated as the
pole
, and
is the angle, in radians, that a ray between the pole and the point makes with a ray
designated as the
polar axis
.
The oordinates
r
and
are alled
polar oordinates
.
The pole is the point (0
;
0) in Cartesian oordinates, and has p olar o ordinates (0
;
) for
any
value of
. The polar axis orresp onds to the positive
x
-axis. An angle
is onsidered positive if
measured in the ounterlokwise diretion from the polar axis, and negative if measured in the
lokwise diretion.
Using these onventions, the Cartesian oordinates of a point an easily be obtained from the
polar o ordinates using the relations
x
=
r
os
; y
=
r
sin
:
(1)
Sine sin
and os
are not one-to-one, and sine
r
is allowed to assume negative values, it follows
that eah point in the plane has innitely many representations in polar oordinates.
The polar oordinates of a point an b e obtained from the Cartesian oordinates as follows:
r
=
x
2
+
y
2
;
tan
=
y
x
:
(2)
It should be noted that b eause tan
is not one-to-one on the interval 0
<
2
, it is neessary
to onsider the signs of
x
and
y
in order to make sure that the proper value of
is used to represent
the p oint (
x; y
). Otherwise, the p oint (
r;
) may lie in the wrong quadrant of the plane.
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