POLAR EQUATIONS
Section 10-4
Polar Coordinates
Given: P(r , )
r: Directed distance from the Polar axis
(pole) to point P
Ɵ: Directed angle from the Polar axis to
ray OP
O
Initial ray
Polar Coordinates
Each point P in the plane can be assigned
polar coordinates (r, Ɵ), as follows.
r = directed distance from O to P
Ɵ = directed angle, counterclockwise from polar
axis to segment OP
Polar Graphs
1) Graph the following
polar coordinates:
Polar Coordinates
In general, the point (r, Ɵ) can be written as
(r, Ɵ) = (r, Ɵ + 2nπ)
or
(r, Ɵ) = (–r, Ɵ + (2n + 1)π)
where n is any integer. Moreover, the pole is
represented by (0, Ɵ), where Ɵ is any angle.
Coordinate Conversion
To establish the relationship between polar and
rectangular coordinates, let the polar axis coincide with
the positive x-axis and the pole with the origin
Because (x, y) lies on a circle of
radius r, it follows that r2 = x2 + y2.
cos  
so x 
sin  
so y 
x2  y2  r 2
so r 
2) Convert
 5 
 2, 
 6 
to rectangular coordinates
3) Convert  2, 2 3  to rectangular coordinates
4) Convert 3,-3 to Polar coordinates
5) Convert the polar equation r cos   4
to rectangular form
4
6) Convert the polar equation r 
2 cos   sin 
to rectangular form
7) Convert the rectangular equation x  y  3
to polar form
8) Convert the rectangular equation xy  2
to polar form
Polar Graphs
The graph of r = a is a
circle of radius a centered
at zero
Ɵ = α is a Line through O
making angle α with the
initial ray
Symmetry
• Symmetric about the x-axis: if the point
(r, Ɵ) lies on the graph, the point (-r, Ɵ)
or (r, π-Ɵ) lies on the graph
• Symmetric about the y-axis: if the point
(r, Ɵ) lies on the graph, the point (-r, -Ɵ)
or (r, π-Ɵ) lies on the graph
• Symmetric about the origin: if the point
(r, Ɵ) lies on the graph, the point (-r, Ɵ)
or (r, π+Ɵ) lies on the graph
Symmetry
Polar Graphs
9) Graph r  5  4 sin  without a graphing
calculator with the values of Ɵ from 0 to 2π.
This curve is called a cardioid.
To plot points use
x  r cos  and y  r sin 
Special Polar Graphs
The following are simpler in polar form than in
rectangular form. The polar equation of a
circle having a radius of a and centered at the
origin is simply r  a r  a  b sin  r  a  b cos 
Special Polar Graphs
Special Polar Graphs
Spiral of Archimedes
1
r 
2
Slope and Tangent Lines
x  r cos   f ( ) cos 
y  r sin   f ( ) sin 
Using the parametric form of dy/dx we have
dr
sin 
 r cos 
dy
d
 d 
dr
dx dx
 r sin 
d cos 
d
dy
Horizontal and Vertical Tangent Lines
• Horizontal
dy
dx
 0 where
0
d
d
• Vertical
dx
dy
 0 where
0
d
d
Cusp at (0, 0)
Tangent Lines at the Pole
If f ( )  0 where f ' ( )  0
dr
dr
 r cos  sin 
0
dy
d
d


 tan 
dx cos  dr  r sin  cos  dr  0
d
d
sin 
then
Then the line   
Is tangent to the pole to
the graph of r  f  
tangents at the pole


6
,

2
,
5
6
10) Find the equation of the line tangent to the
polar curve
3
r  sin 2 , when  
4
11) Find the vertical and horizontal tangents fo
r  1  cos  , 0    2
HOME WORK
Worksheet 10-4