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Section 8.2 Equations and Graphs
In this section, we cover the standard polar graphs. Most of these graphs have simple polar equations,
but the graphs would be diﬃcult to deﬁne in cartesian coordinates.
Note: As with cartesian equations, when one plots a polar equation, e.g.
r
=s
in
θ
, one plots the set
of all points (
r, θ
) that satisfy the equation.
Example.
The graph of
r
= 2 is a circle of radius 2 centered at the origin, because we plot the
set of all points (
r, θ
) such that
r
= 2. Note that most circles not centered at the origin have fairly
complicated polar equations.
Example.
The graph of
θ
=
π
3
is a line through the origin with slope
√
3. This is the graph of all
(
r, θ
) such that
θ
=
π
3
Example.
The graph of
r
cos
θ
= 1 is the vertical line
x
= 1, because
r
cos
θ
=
x
and the polar
equation
r
cos
θ
= 1 is equivalent to the cartesian equation
x
=1.
Example.
Graph
r
=2sin
θ
.
Solution. We ﬁnd the cartesian equation for
r
=2s
θ
.
r
2
=2
r
sin
θ
(multiply by
r
)
x
2
+
y
2
y
(
r
2
=
x
2
+
y
2
,
y
=
r
sin
θ
)
x
2
+
y
2

2
y
=0
x
2
+(
y

1)
2
=1
This last equation is the equation of a circle of radius 1 tangent to the
x

axis as shown below.
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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.
 Fall '11
 Kutter
 Algebra, Equations

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