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Polar Coordinates:
There is an alternative method for locating points in the plane called polar
coordinates.
x
y
( )x,y
( )r ,
r
Rectangular/Cartesian coordinates are unique. Polar coordinates are not unique.
Example:
1. Find polar coordinates for the point with rectangular coordinates ( )11, .
2, , 2, 2 , 2, 4 ,4 4 4
5 5 52, , 2, 2 , 2, 4 ,
4 4 4
− − −
Yes, there are infinitely many polar coordinates for a given pair of rectangular
coordinates. Infinitely many angles, and r can also be negative.
2. Find polar coordinates for the point with rectangular coordinates ( )0 0, .
( )0, , where is any angle. Even the origin has infinitely many polar coordinates.
1
1 2
4
5
4
Conversion Equations:
2 2 2x y r+ =
x r cos=
y r sin=
ytan
x =
Examples:
1. Find polar coordinates for the rectangular coordinates ( )3 1,− .
( )2
2 2 2 2 2 23 1 4
1
3
5 5So I'll use 2 and to get 2
6 6
x y r r r
ytan tan
x
r , .
+ = − + = =
= = −
= =
2. Find rectangular coordinates for the polar coordinates 4
43
,
.
( )
44 2
3
44 2 3
3
So I'll get 2 2 3
x r cos x cos x
y r sin y sin y
, .
= = = −
= = = −
− −
3. Transform the rectangular coordinate equation 2 2 4x y+ = into an equivalent polar
coordinate equation, and graph the solution curve.
2 2 24 4 2 or 2x y r r r+ = = = = −
2
4. Transform the rectangular coordinate equation 3x = into an equivalent polar
coordinate equation, and graph the solution curve.
33 3 3x r cos r r sec
cos
= = = =
5. Transform the rectangular coordinate equation 2y x= into an equivalent polar
coordinate equation, and graph the solution curve.
( )22 2y x r sin r cos sin r cos r sec tan = = = =
3
6. Transform the polar coordinate equation 2r sin = into an equivalent rectangular
coordinate equation, and graph the solution curve.
2 2r sin y = =
7. Transform the polar coordinate equation 2r cos= into an equivalent rectangular
coordinate equation, and graph the solution curve.
( )22 2 2 2 2 22 2 2 2 0 1 1r cos r r cos x y x x x y x y = = + = − + = − + =
2
2
8. Transform the polar coordinate equation r sin cos = − into an equivalent
rectangular coordinate equation, and graph the solution curve.
( ) ( )2 22 2 2 2 2 1 1
2 2
10
2r sin cos r r sin r cos x y y x x x y y x y = − = − + = − + + − = + + − =
1
-1
Special Polar Coordinate Equations/Graphs:
Cardioid:
1. 1r cos= +
Start with an r vs. plot, treating r and as rectangular coordinates.
Use the r vs. plot to help you create
The polar coordinate graph.
0 2
3
2
2
2
1
0,2 =
2
=
=
3
2
=
2
1
1−
r
If r is zero for a particular value
0 , then the graph comes into the
origin tangent to the line 0
= .
Here, the value is .
2. ( )2 1r sin= −
0
2
3
2
2
2
4
0,2 =
2
=
=
3
2
=
2 2−
4−
r
To find the angle(s) where r is zero, solve
the trig. equation, ( )2 1 0sin− = .
Cardioid with an Inner Loop:
1. 1 2r cos= +
3
0 2
3
4
3
2
3
1
1−
1 0,2 =
2
3
=
4
3
=
=
The inner loop is traced
when r takes on negative
values for between 2
3
and 4
3
. The graph
approaches the origin
tangent to these two lines.
r
To find the angle(s) where r is zero,
solve the trig. equation, 1 2 0cos+ = .
2. 1 2r sin= +
1
1−
3
0
2
7
6
11
6
3
2
2
r
1
1 1−
3
0,2 =
2
=
=
7
6
=
11
6
=
3
2
=
The inner loop is traced
when r takes on negative
values for between 7
6
and 11
6
. The graph
approaches the origin
tangent to these two lines.
To find the angle(s) where r is zero,
solve the trig. equation, 1 2 0sin+ = .
Rose:
( )2 2r sin =
To find the angle(s) where r is zero,
solve the trig. equation, 2 2 0sin = .
2
2−
0
4
2
3
4
5
4
3
2
7
4
2
r
0,2 =
2
=
=
3
2
=
4
=
3
4
=
5
4
=
7
4
=
The tips of the four petals occur
for 3 5 7
, , ,4 4 4 4
= . The graph
approaches the origin tangent to
these lines corresponding to the
angles 3
0, , ,2 2
= . Two of the
petals are traced with positive r
values and two with negative r
values.
Lemniscate:
( )2 2r cos =
Start with an 2r vs. plot, and then
convert it into an r vs. plot by
taking square-roots. Note that 2r can’t
be negative.
1
4
3
4
5
4
7
4
2
0
2r
4
3
4
5
4
7
4
2 0
r
1 1−
4
=
3
4
=
5
4
=
7
4
=
The positive r values trace the entire curve once, and the negative r values
trace it again.
Spiral:
0r ; =
2
−
3
2
−
2
Intersections of Polar Graphs:
1. Find the points of intersection of the graphs of the polar coordinate equations 1r =
and 1r cos= − .
Set the equations equal to each other, solve,
and use symmetry in the graph, if possible.
31 cos 1 cos 0 , ,
2 2
So the two points of intersection are
31, and 1, .
2 2
− = = =
2. Find the points of intersection of the graphs of the polar coordinate equations 1r =
and ( )2 2r sin = .
Set the equations equal to each other, solve,
and use symmetry in the graph, if possible.
12sin 2 1 sin 2 2 , 2 ,
2 6 6
5 5or 2 , 2 , , ,
6 6 12 12
5 5or , ,
12 12
So from symmetry, the eigh
= = = +
= + = +
= +
t points of intersection are
5 7 11 13 17 19 231, , 1, , 1, , 1, , 1, , 1, , 1, , 1, .
12 12 12 12 12 12 12 12
3. Find the points of intersection of the graphs of the polar coordinate equations
1r cos= − and r cos= .
( )
Set the equations equal to each other, solve,
and use symmetry in the graph, if possible.
1cos 1 cos cos , 2 ,
2 3 3
5 5or , 2 ,
3 3
So from the graph, the three points of intersection are
10,0 ,
= − = = +
= +
1 5, , , .
2 3 2 3