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