UNIT 3
POLAR COORDINATES, GRAPHING
INTRO: We will continue our study of polar coordinates. The emphasis will be on graphing.
Here are the transformations between Cartesian coordinates and polar coordinates introduced
in the last Unit.
r
=
p
x
2
+
y
2
x
=
r
cos
θ
tan
θ
=
y
x
y
=
r
sin
θ
1. Graphs of Polar Equations
We wish to plot a polar equation of the form
r
=
f
(
θ
), where the right hand side is a function
of sin
nθ
or cos
nθ
for n=1,2,3
. . . .
Rather than plotting many points for different
θ
’s, we will
outline a much better method to plot simple polar equations. We first explain the method and
then try it on the following four examples:
r
= cos
θ
r
= 3 sin 2
θ
r
= 2 cos 3
θ
r
= 1 + 2 sin 2
θ
However, before beginning the plot, we must explain a peculiar convention.
In polar equations and inequalities,
r
is permitted to be negative – as we will see in the
examples – even though negative
r
does not make a lot of sense, since
r
=
p
x
2
+
y
2
is always
positive. Here is the convention: if, for a particular value of
θ
,
r
is negative, then the point
to be plotted is rotated 180 degrees. For example, if, for
θ
=
π/
6,
r
=

2, then this really
means
r
= 2 but at the angle
θ
=
π/
6 +
π
= 7
π/
6. This convention is usually followed only in
equations and inequalities. It would be very abnormal, for example, to refer to the Cartesian
point (
x, y
) = (1
,
0), which is
r
= 1
, θ
= 0, as
r
=

1 with
θ
=
π
.
Now for the plotting of a polar equation of the form
r
=
f
(
θ
).
If the right hand side of the equation for
r
contains sin
θ
, we place the values of sin
θ
at the ends of the axes (We may change these labels very shortly): 0 at the end of the
positive
x
axis (since sin 0 = 0), 1 at the end of the positive
y
axis, then 0 and

1 at the
ends of the negative axes. Likewise, 1,0,

1,0 would be written on the axes if the equation
contained cos
θ
(going around counterclockwise, of course, since this is the direction of
increasing
θ
, and beginning with 1 since cos 0 = 1). If, on the other hand, the equation
contained sin 2
θ
or cos 2
θ
, then then we first draw dotted halfaxes at 45 degrees in each
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quadrant, and label all the axes with the value of the trig function, 0,1,0,

1,0,1,0,

1 on
the axes in the case of sin 2
θ
and 1,0,

1,0,1,0,

1,0 in the case of cos 2
θ
. Similarly for
the case of sin 3
θ
and cos 3
θ
.
The axes for this first step are shown below for various
possibilities.
0
0
1
Minus
1
1
1
Minus
1
Minus
1
0
0
0
0
0
0
1
Minus
1
1
Minus
1
0
0
0
1
Minus
1
0
r
a function of cos
θ
r
a function of sin 2
θ
r
a function of cos 3
θ
Using these values of the trig function, we then use the equation for
r
to change these
values to the actual values of
r
at the angles of these axes. Having the values of
r
already
on the graph at the correct angles makes it much easier to evaluate
r
as we sweep around
the origin. You can see these axes labeled with the actual values of
r
rather than the
values of sin
θ
and cos
θ
in the graphs of Figs.
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 Fall '08
 DONTREMEMBER
 Geometry, Transformations, Cartesian Coordinate System, Polar Coordinates, Cos, Polar coordinate system, θ

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