10.2 Graphing Polar Equations
OBJECTIVE 1: Sketching Equations of the Form r cos   a , r sin   a , ar cos   br sin   c , and   
Graphs of Polar Equations of the Form r cos   a , r sin   a , ar cos   br sin   c , and   
where a, b, and c are constants.
The graph of r cos   a is a vertical line.
The graph of ra cos   br sin   c is a line
a
c
with slope m   and y  intercept .
b
b
The graph of r sin   a is a horizontal line.
The graph of    is a line through the pole that
makes an angle of  with the polar axis.
OBJECTIVE 2: Sketching Equations of the Form r  a , r  a sin , and r  a cos 
Graphs of Polar Equations of the Form r  a , r  a sin  , and r  a cos  where a  0 is a constant
The graph of r  a cos  is a circle
The graph of r  a is a circle
centered at the pole with
radius of length a .
a
units
2
a
from the pole with radius of length
.
The graph of r  a sin  is a circle
2
a

centered on the line = that is
units
2
2
a
from the pole with radius of length
.
2
centered on the polar axis that is
OBJECTIVE 3: Sketching Equations of the Form r  a  b sin  and r  a  b cos 
Graphs of Polar Equations of the Form r  a  b sin  and r  a  b cos  where a  0 and b  0 are
constants.
The graph is a cardioid if
a
 1.
b
The graph is a limacon with a dimple if 1<
The graph is a limacon with an inner loop if
a
<1.
b
a
<2.
b
The graph is a limacon with no inner loop and no dimple if
a
 2.
b
Steps for Sketching Polar Equations (Limacons) of the Form r  a  b sin  and r  a  b cos 
Step 1. Identify the general shape using the ratio
a
.
b
a
 1 , then the graph is a cardioid.
b
a
 1 , then the graph is a limacon with an inner loop that intersects the pole.
•If
b
a
•If 1   2 , then the graph is a limacon with a dimple .
b
a
 2 , then the graph is a limacon with no inner loop and no dimple.
.•If
b
•If
Step 2. Determine the symmetry.

.
2
•If the equation is of the form r  a  b cos  , then the graph must be symmetric about the polar axis.
•If the equation is of the form r  a  b sin  , then the graph must be symmetric about the line  
Step 3. Plot the points corresponding to the quadrantal angles   0 ,  

3
,    , and  
.
2
2
Step 4. If necessary, plot a few more points until symmetry can be used to complete the graph.
OBJECTIVE 4: Sketching Equations of the Form r  a sin n and r  a cos n
Graphs of Polar Equations of the Form r  a sin n and r  a cos n where a  0 is a constant and
n  1 is a positive integer.
The graph is a rose with n petals.
The endpoint of one petal lies

3
along the vertical line = or = .
2
2
The graph is a rose with 2n petals.
None of the petals have an endpoint
lying on either the polar axis or the

line = .
2
The graph is a rose with n petals.
The endpoint of one petal lies
along the polar axis.
The graph is a rose with 2n petals.

3
on the lines = and = . Two petals
2
2
have endpoints that lie on the polar axis.
Steps for Sketching Polar Equations (Roses) of the Form r  a sin n and r  a cos n where
a  0 and n  1is a positive integer.
Step 1. Identify the number of “petals”.
•If n is even, then there are 2n petals.
•If n is odd, then there are n petals.
Step 2. Determine the length of each petal.
•The length of each petal is a units.
Step 3. Determine all angles where an endpoint of a petal lies.
•If the equation is of the form r  a sin n , then the endpoints occur for angles on the interval 0, 2  *
that satisfy the equations sin n  1 and sin n  1 .
•If the equation is of the form r  a cos n , then the endpoints occur for angles on the interval 0, 2  *
that satisfy the equations cos n  1 and cos n  1 .
*Note that when n is odd, it is only necessary to consider angles on the interval 0,   . A complete
graph is obtained on this interval because the graph will completely traverse itself on the interval
 , 2  .
Step 4. Substitute each angle determined in Step 3 back into the original equation to obtain the appropriate
values of r for each angle. The ordered pairs obtained represent the endpoints of the rose petals.
Plot these points on the graph.
Step 5. Determine angles where the graph passes through the pole. These angles serve as a guide when
sketching the width of a petal.
•If the equation is of the form r  a sin n , then the graph passes through the pole when sin n  0 .
•If the equation is of the form r  a cos n , then the graph passes through the pole when cos n  0 .
Step 6. Draw each petal to complete the graph.
OBJECTIVE 5: Sketching Equations of the Form r 2  a 2 sin 2 and r 2  a 2 cos 2
Graphs of Polar Equations of the Form r 2  a 2 sin 2 and r 2  a 2 cos 2 where a  0 is a constant
The graph is a lemniscate symmetric about the
pole with the endpoints of the two loops lying

5
along the angles   and  
.
4
4
The graph is a lemniscate symmetric about the

pole, the polar axis, and the vertical line = .
The endpoints of the two loops lie along the 2
angles   0and   .