The document discusses polar coordinates and graphs. It begins by explaining how polar coordinates (r, θ) track the location of a point P in the plane, where r is the distance from the origin and θ is the angle from the positive x-axis. It then provides the conversions between rectangular (x, y) and polar coordinates. The document introduces polar equations, showing how they relate r and θ, and gives examples of the constant equations r = c and θ = c, which describe a circle and line, respectively. It concludes by explaining how to graph other polar equations by plotting points using a polar graph paper.

1. Polar Coordinates & Graphs

2. Polar Coordinates & Graphs
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ),
P
x
y
(r, )p

3. Polar Coordinates & Graphs
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
P
x
y
(r, )p
r

4. Polar Coordinates & Graphs
P
x
y
O
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
and  = a is an angle that is measured from the
positive x–axis which gives the direction to P.
(r, )p
r

5. Polar Coordinates & Graphs
Conversion Rules
Let (x, y)R and (r, )P be the
rectangular and the polar
coordinates of P,
P
x
y
O
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
and  = a is an angle that is measured from the
positive x–axis which gives the direction to P.
(r, )p = (x, y)R
r

6. Polar Coordinates & Graphs
Conversion Rules
Let (x, y)R and (r, )P be the
rectangular and the polar
coordinates of P, then
x = r*cos()
P
x
y
O
x = r*cos()
The rectangular and polar conversions
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
and  = a is an angle that is measured from the
positive x–axis which gives the direction to P.
(r, )p = (x, y)R
r

7. Polar Coordinates & Graphs
Conversion Rules
Let (x, y)R and (r, )P be the
rectangular and the polar
coordinates of P, then
x = r*cos()
y = r*sin()
P
x
y
O
x = r*cos()
y = r*sin()
The rectangular and polar conversions
r
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
and  = a is an angle that is measured from the
positive x–axis which gives the direction to P.
(r, )p = (x, y)R

8. Polar Coordinates & Graphs
Conversion Rules
Let (x, y)R and (r, )P be the
rectangular and the polar
coordinates of P, then
x = r*cos()
y = r*sin()
r = √x2 + y2
tan() = y/x
P
x
y
O
x = r*cos()
y = r*sin()
The rectangular and polar conversions
r = √x2 + y2
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
and  = a is an angle that is measured from the
positive x–axis which gives the direction to P.
(r, )p = (x, y)R

9. Polar Coordinates & Graphs
In the last section, we tracked the location of a point
P in the plane by its polar coordinates (r, ), where
r = a is the signed distance from the origin (0, 0) to P,
and  = a is an angle that is measured from the
positive x–axis which gives the direction to P.
Conversion Rules
Let (x, y)R and (r, )P be the
rectangular and the polar
coordinates of P, then
P
x
y
O
x = r*cos()
y = r*sin()
The rectangular and polar conversions
x = r*cos()
y = r*sin()
r = √x2 + y2
tan() = y/x
r = √x2 + y2
If P is in quadrants I, II or IV
then  may be extracted
by inverse trig functions. But if P is in quadrant III, then
 can’t be calculated directly by inverse trig-functions.
(r, )p = (x, y)R

10. Polar Equations
Polar Coordinates & Graphs

11. Polar Equations
A rectangular equation in x and y gives the relation
between the horizontal displacement x and vertical
displacement y of locations.
Polar Coordinates & Graphs

12. Polar Equations
A rectangular equation in x and y gives the relation
between the horizontal displacement x and vertical
displacement y of locations. A polar equation gives a
relation of the distance r the direction .
Polar Coordinates & Graphs

13. Polar Equations
A rectangular equation in x and y gives the relation
between the horizontal displacement x and vertical
displacement y of locations. A polar equation gives a
relation of the distance r the direction .
Polar Coordinates & Graphs
The rectangular equation y = x
specifies that the horizontal
displacement x must be the same
as the vertical displacement y for
our points P.

14. Polar Equations
A rectangular equation in x and y gives the relation
between the horizontal displacement x and vertical
displacement y of locations. A polar equation gives a
relation of the distance r the direction .
Polar Coordinates & Graphs
The rectangular equation y = x
specifies that the horizontal
displacement x must be the same
as the vertical displacement y for
our points P.
y
y
x
x
P(x, y)
The graph of y = x in the
the rectangular system

15. Polar Equations
A rectangular equation in x and y gives the relation
between the horizontal displacement x and vertical
displacement y of locations. A polar equation gives a
relation of the distance r the direction .
Polar Coordinates & Graphs
The rectangular equation y = x
specifies that the horizontal
displacement x must be the same
as the vertical displacement y for
our points P.
y
y
x
x
P(x, y)
The graph of y = x in the
the rectangular system
The polar equation r = rad says that
the distance r must be the same as
the rotational measurement  for P.
Its graph is the Archimedean spiral.

16. Polar Equations
A rectangular equation in x and y gives the relation
between the horizontal displacement x and vertical
displacement y of locations. A polar equation gives a
relation of the distance r the direction .
Polar Coordinates & Graphs
The rectangular equation y = x
specifies that the horizontal
displacement x must be the same
as the vertical displacement y for
our points P.
y
y
x
x
P(x, y)
x
P(r, )

r
The graph of y = x in the
the rectangular system
Graph of r =  in the
polar system.
The polar equation r = rad says that
the distance r must be the same as
the rotational measurement  for P.
Its graph is the Archimedean spiral.

17. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c

18. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c)
b. ( = c)

19. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c) The constant equation r = c
indicates that “the distance r is c, a
fixed constant” and that  may be of
any value.
b. ( = c)

20. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c) The constant equation r = c
indicates that “the distance r is c, a
fixed constant” and that  may be of
any value. This equation describes the
circle of radius c, centered at (0,0).
b. ( = c)

21. Polar Coordinates
Example A. a. Plot the following polar coordinates
A(4, 60o)P , B(5, 0o)P, C(4, –45o)P, D(–4, 3π/4 rad)P.
Find their corresponding rectangular coordinates.
For A(4, 60o)P
x = r*cos()
y = r*sin()
(x, y)R = (4*cos(60⁰), 4*sin(60⁰)),
r2 = x2 + y2
tan() = y/x
x
y
60o
4
A(4, 60o)P

22. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c) The constant equation r = c
indicates that “the distance r is c, a
fixed constant” and that  may be of
any value. This equation describes the
circle of radius c, centered at (0,0).
x
y
c
The constant
equation r = c
b. ( = c)

23. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c) The constant equation r = c
indicates that “the distance r is c, a
fixed constant” and that  may be of
any value. This equation describes the
circle of radius c, centered at (0,0).
x
y
c
The constant
equation r = c
b. ( = c) The constant equation  = c
requires that “the directional angle  is c,
a fixed constant” and the distance r may
be of any value.

24. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c) The constant equation r = c
indicates that “the distance r is c, a
fixed constant” and that  may be of
any value. This equation describes the
circle of radius c, centered at (0,0).
x
y
c
The constant
equation r = c
b. ( = c) The constant equation  = c
requires that “the directional angle  is c,
a fixed constant” and the distance r may
be of any value. This equation describes
the line with polar angle c.

25. Let’s look at some basic examples of polar graphs.
Polar Coordinates & Graphs
The Constant Equations r = c &  = c
Example A. Graph the following polar
equations.
a. (r = c) The constant equation r = c
indicates that “the distance r is c, a
fixed constant” and that  may be of
any value. This equation describes the
circle of radius c, centered at (0,0).
x
y
c
The constant
equation r = c
b. ( = c) The constant equation  = c
requires that “the directional angle  is c,
a fixed constant” and the distance r may
be of any value. This equation describes
the line with polar angle c.
x
y
The constant
equation  = c
 = c

26. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly.

27. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.

28. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos().

29. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o.

30. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o.

31. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o.

32. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o.

33. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o.

34. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o. 3

35. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o. 3

36. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o. 3

37. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o. 3

38. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
Let’s plot the
points starting
with  going
from 0 to 90o. 3

39. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
–1.5 120o
≈ –2.1 135o
≈ –2.6 150o
–3 180o
Let’s plot the
points starting
with  going
from 0 to 90o. 3
Next continue
with  from
90o to 180o as
shown in the table.

40. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
–1.5 120o
≈ –2.1 135o
≈ –2.6 150o
–3 180o
Let’s plot the
points starting
with  going
from 0 to 90o. 3
Next continue
with  from
90o to 180o as
shown in the table.

41. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
–1.5 120o
≈ –2.1 135o
≈ –2.6 150o
–3 180o
Let’s plot the
points starting
with  going
from 0 to 90o. 3
Next continue
with  from
90o to 180o as
shown in the table.
negative so the points are in the 4th quadrant.
Note the r’s are

42. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly. For this we will use the polar graph
paper such as the one shown here which is gridded in
terms of the distances and directional angles.
Example B. a. Graph r = 3cos(). r 
3 0o
3(3/2) ≈ 2.6 30o
3(2/2) ≈ 2.1 45o
1.5 60o
0 90o
–1.5 120o
≈ –2.1 135o
≈ –2.6 150o
–3 180o
Let’s plot the
points starting
with  going
from 0 to 90o. 3
Next continue
with  from
90o to 180o as
shown in the table.
negative so the points are in the 4th quadrant.
Note the r’s are

43. Polar Coordinates & Graphs
Continuing with  from 180o to 270o, r 
–3 180o
≈ –2.6 210o
≈ –2.1 225o
–1.5 240o
0 270o
1.5 300o
≈ 2.1 315o
≈ 2.6 330o
3 360o
again r’s are
negative, hence
the points are
located in the
1st quadrant.

44. Polar Coordinates & Graphs
Continuing with  from 180o to 270o, r 
–3 180o
≈ –2.6 210o
≈ –2.1 225o
–1.5 240o
0 270o
1.5 300o
≈ 2.1 315o
≈ 2.6 330o
3 360o
again r’s are
negative, hence
the points are
located in the
1st quadrant.
In fact, they
trace over the
same points as  goes from 0o to 90o

45. Polar Coordinates & Graphs
Continuing with  from 180o to 270o, r 
–3 180o
≈ –2.6 210o
≈ –2.1 225o
–1.5 240o
0 270o
1.5 300o
≈ 2.1 315o
≈ 2.6 330o
3 360o
again r’s are
negative, hence
the points are
located in the
1st quadrant.
3
In fact, they
trace over the
same points as  goes from 0o to 90o

46. Polar Coordinates & Graphs
Continuing with  from 180o to 270o, r 
–3 180o
≈ –2.6 210o
≈ –2.1 225o
–1.5 240o
0 270o
1.5 300o
≈ 2.1 315o
≈ 2.6 330o
3 360o
again r’s are
negative, hence
the points are
located in the
1st quadrant.
3
In fact, they
trace over the
Finally as  goes from 270o to 360o we trace over the
same points as  goes from 90o to 180o in the 4th
quadrant.
same points as  goes from 0o to 90o

47. Polar Coordinates & Graphs
Continuing with  from 180o to 270o, r 
–3 180o
≈ –2.6 210o
≈ –2.1 225o
–1.5 240o
0 270o
1.5 300o
≈ 2.1 315o
≈ 2.6 330o
3 360o
again r’s are
negative, hence
the points are
located in the
1st quadrant.
3
In fact, they
trace over the
Finally as  goes from 270o to 360o we trace over the
same points as  goes from 90o to 180o in the 4th
quadrant. As we will see shortly, these points form a
circle and for every period of 180o the graph of
r = cos() traverses this circle once.
same points as  goes from 0o to 90o

48. Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
3

49. Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
Multiply both sides by r so we
have r2 = 3r*cos(). 3

50. Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
Multiply both sides by r so we
have r2 = 3r*cos().
In terms of x and y, it’s
x2 + y2 = 3x
3

51. Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
Multiply both sides by r so we
have r2 = 3r*cos().
In terms of x and y, it’s
x2 + y2 = 3x
x2 – 3x + y2 = 0
3

52. Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
Multiply both sides by r so we
have r2 = 3r*cos().
In terms of x and y, it’s
x2 + y2 = 3x
x2 – 3x + y2 = 0
completing the square,
x2 – 3x + (3/2)2 + y2 = 0 + (3/2)2
3

53. Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
Multiply both sides by r so we
have r2 = 3r*cos().
In terms of x and y, it’s
x2 + y2 = 3x
x2 – 3x + y2 = 0
completing the square,
x2 – 3x + (3/2)2 + y2 = 0 + (3/2)2
(x – 3/2)2 + y2 = (3/2)2
3

54. 3
Polar Coordinates & Graphs
b. Convert r = 3cos() to a
rectangular equation. Verify it’s
a circle and find the center and
radius of this circle.
Multiply both sides by r so we
have r2 = 3r*cos().
In terms of x and y, it’s
x2 + y2 = 3x
x2 – 3x + y2 = 0
completing the square,
x2 – 3x + (3/2)2 + y2 = 0 + (3/2)2
(x – 3/2)2 + y2 = (3/2)2
so the points form the circle centered at (3/2, 0)
with radius 3/2.

55. Polar Coordinates & Graphs
In general, the polar
equations of the form
r = ±D*cos()
r = ±D*sin()
are circles with diameter D
and tangent to the x or y axis
at the origin. r = ±a*cos()
r = ±a*sin()
D x
y

56. Polar Coordinates & Graphs
In general, the polar
equations of the form
r = ±D*cos()
r = ±D*sin()
are circles with diameter D
and tangent to the x or y axis
at the origin. r = ±a*cos()
r = ±a*sin()
D x
y
The points (r, ) and (r, –) are the
vertical mirror images of each other
across the x–axis.
x
(r, )
(r, –)
1

57. Polar Coordinates & Graphs
In general, the polar
equations of the form
r = ±D*cos()
r = ±D*sin()
are circles with diameter D
and tangent to the x or y axis
at the origin. r = ±a*cos()
r = ±a*sin()
D x
y
The points (r, ) and (r, –) are the
vertical mirror images of each other
across the x–axis. So if r = f() = f(–)
such as r = cos() = cos(–),
then its graph is symmetric with
respect to the x–axis,
x
(r, )
(r, –)
1

58. Polar Coordinates & Graphs
In general, the polar
equations of the form
r = ±D*cos()
r = ±D*sin()
are circles with diameter D
and tangent to the x or y axis
at the origin. r = ±a*cos()
r = ±a*sin()
D x
y
The points (r, ) and (r, –) are the
vertical mirror images of each other
across the x–axis. So if r = f() = f(–)
such as r = cos() = cos(–),
then its graph is symmetric with
respect to the x–axis,
x
(r, )
(r, –)
r = cos() = cos(–)
1

59. Polar Coordinates & Graphs
In general, the polar
equations of the form
r = ±D*cos()
r = ±D*sin()
are circles with diameter D
and tangent to the x or y axis
at the origin. r = ±a*cos()
r = ±a*sin()
D x
y
The points (r, ) and (r, –) are the
vertical mirror images of each other
across the x–axis. So if r = f() = f(–)
such as r = cos() = cos(–),
then its graph is symmetric with
respect to the x–axis, so r = ±D*cos()
are the horizontal circles.
x
(r, )
(r, –)
r = cos() = cos(–)
1

60. Polar Coordinates & Graphs
x
(r, )(–r, –)
y
The points (r, ) and (–r, –) are the
mirror images of each other across
the y–axis.

61. Polar Coordinates & Graphs
x
(r, )
r = sin() = –sin(–)
(–r, –)
y
The points (r, ) and (–r, –) are the
mirror images of each other across
the y–axis. So if r = f(–) = –f()
such as r = sin() = –sin(–),
then its graph is symmetric to the
y–axis and so r = ±D*sin()
are the two vertical circles.

62. Polar Coordinates & Graphs
x
(r, )
r = sin() = –sin(–)
(–r, –)
y
x
y
r = cos()r = –cos()
r = sin()
r = –sin()
1
1
Here they are with their orientation
starting at  = 0.
The points (r, ) and (–r, –) are the
mirror images of each other across
the y–axis. So if r = f(–) = –f()
such as r = sin() = –sin(–),
then its graph is symmetric to the
y–axis and so r = ±D*sin()
are the two vertical circles.

63. Polar Coordinates & Graphs
The Cardioids
r = c(1 ± cos())
r = c(1 ± sin())
The graphs of the equations of the form
are called the cardioids, or the heart shaped curves.
Example C. Graph r = 1 – cos().
The graph of r = 1 – cos() is symmetric with respect
to the x–axis because cos() = cos(–). Therefore we
will plot  from 0o to 180o and take its mirrored image
across the x–axis for the complete graph. As  goes
from 0o to 180o, cos() goes from 1 to –1, and the
expression 1 – cos() goes from 0 to 2. The table is
shown below, readers may verify the approximate
values of r’s.

64. Polar Coordinates & Graphs
The Cardioids
r = c(1 ± cos())
r = c(1 ± sin())
The graphs of the equations of the form
are called the cardioids, or the heart shaped curves.
Example C. Graph r = 1 – cos().

65. Polar Coordinates & Graphs
The Cardioids
r = c(1 ± cos())
r = c(1 ± sin())
The graphs of the equations of the form
are called the cardioids, or the heart shaped curves.
Example C. Graph r = 1 – cos().
The graph of r = 1 – cos() is symmetric with respect
to the x–axis because cos() = cos(–).

66. Polar Coordinates & Graphs
The Cardioids
r = c(1 ± cos())
r = c(1 ± sin())
The graphs of the equations of the form
are called the cardioids, or the heart shaped curves.
Example C. Graph r = 1 – cos().
The graph of r = 1 – cos() is symmetric with respect
to the x–axis because cos() = cos(–). Therefore we
will plot  from 0o to 180o and take its mirrored image
across the x–axis for the complete graph.

67. Polar Coordinates & Graphs
The Cardioids
r = c(1 ± cos())
r = c(1 ± sin())
The graphs of the equations of the form
are called the cardioids, or the heart shaped curves.
Example C. Graph r = 1 – cos().
The graph of r = 1 – cos() is symmetric with respect
to the x–axis because cos() = cos(–). Therefore we
will plot  from 0o to 180o and take its mirrored image
across the x–axis for the complete graph. As  goes
from 0o to 180o, cos() goes from 1 to –1, and the
expression 1 – cos() goes from 0 to 2.

68. Polar Coordinates & Graphs
The Cardioids
r = c(1 ± cos())
r = c(1 ± sin())
The graphs of the equations of the form
are called the cardioids, or the heart shaped curves.
Example C. Graph r = 1 – cos().
The graph of r = 1 – cos() is symmetric with respect
to the x–axis because cos() = cos(–). Therefore we
will plot  from 0o to 180o and take its mirrored image
across the x–axis for the complete graph. As  goes
from 0o to 180o, cos() goes from 1 to –1, and the
expression 1 – cos() goes from 0 to 2. The table is
shown below, readers may verify the approximate
values of r’s.

69. r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
Polar Coordinates & Graphs
x
2

70. r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
Polar Coordinates & Graphs
x
2

71. r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
Polar Coordinates & Graphs
x
2

72. Polar Coordinates & Graphs
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2

73. Polar Coordinates & Graphs
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
Reflecting across the x–axis, we have the cardioid.
x
2

74. Polar Coordinates & Graphs
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2
Reflecting across the x–axis, we have the cardioid.

75. Polar Coordinates & Graphs
The cardioid is the
track of a point on
a circle as it
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2
Reflecting across the x–axis, we have the cardioid.
revolves around another circle of the same size.

76. Polar Coordinates & Graphs
The cardioid is the
track of a point on
a circle as it
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2
Reflecting across the x–axis, we have the cardioid.
revolves around another circle of the same size.

77. Polar Coordinates & Graphs
The cardioid is the
track of a point on
a circle as it
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2
Reflecting across the x–axis, we have the cardioid.
revolves around another circle of the same size.

78. Polar Coordinates & Graphs
The cardioid is the
track of a point on
a circle as it
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2
Reflecting across the x–axis, we have the cardioid.
revolves around another circle of the same size.

79. Polar Coordinates & Graphs
The cardioid is the
track of a point on
a circle as it
r=1–cos() 
0 0o
≈ 0.13 30o
≈ 0.29 45o
0.5 60o
1 90o
1.5 120o
≈ 1.71 135o
≈ 1.87 150o
2 180o
x
2
Reflecting across the x–axis, we have the cardioid.
revolves around another circle of the same size.

80. Polar Coordinates & Graphs
We use the conversion rules to convert equations
between the rectangular and the polar coordinates.

81. Polar Coordinates & Graphs
We use the conversion rules to convert equations
between the rectangular and the polar coordinates.
Example C. Convert each equation into a
corresponding rectangular form.
a. r = 5

82. Polar Coordinates & Graphs
We use the conversion rules to convert equations
between the rectangular and the polar coordinates.
Example C. Convert each equation into a
corresponding rectangular form.
The polar equation states that the
distance from the origin to the points
on its graph is a constant 5.
a. r = 5

83. Polar Coordinates & Graphs
We use the conversion rules to convert equations
between the rectangular and the polar coordinates.
Example C. Convert each equation into a
corresponding rectangular form.
The polar equation states that the
distance from the origin to the points
on its graph is a constant 5.
This is the circle of radius 5,
centered at (0, 0).
a. r = 5
r = 5 x
y

84. Polar Coordinates & Graphs
We use the conversion rules to convert equations
between the rectangular and the polar coordinates.
Example C. Convert each equation into a
corresponding rectangular form.
The polar equation states that the
distance from the origin to the points
on its graph is a constant 5.
This is the circle of radius 5,
centered at (0, 0).
a. r = 5
r = 5 x
Set r = √x2 + y2 = 5 we have that
x2 + y2 = 52 or the rectangular form of a circle.
y

85. Polar Coordinates & Graphs
b. Convert r = 4cos() into the rectangular form.

86. Polar Coordinates & Graphs
Multiplying by r to both sides, we have
r2 = 4 rcos(),
b. Convert r = 4cos() into the rectangular form.

87. Polar Coordinates & Graphs
Multiplying by r to both sides, we have
r2 = 4 rcos(), in terms of x and y, we have
x2 + y2 = 4x
x2 – 4x + y2 = 0
b. Convert r = 4cos() into the rectangular form.

88. Polar Coordinates & Graphs
Multiplying by r to both sides, we have
r2 = 4 rcos(), in terms of x and y, we have
x2 + y2 = 4x
x2 – 4x + y2 = 0 completing the square,
x2 – 4x + 4 + y2 = 4
b. Convert r = 4cos() into the rectangular form.

89. Polar Coordinates & Graphs
Multiplying by r to both sides, we have
r2 = 4 rcos(), in terms of x and y, we have
x2 + y2 = 4x
x2 – 4x + y2 = 0 completing the square,
x2 – 4x + 4 + y2 = 4 we have
(x – 2)2 + y2 = 4
b. Convert r = 4cos() into the rectangular form.

90. Polar Coordinates & Graphs
2
Multiplying by r to both sides, we have
r2 = 4 rcos(), in terms of x and y, we have
x2 + y2 = 4x
x2 – 4x + y2 = 0 completing the square,
x2 – 4x + 4 + y2 = 4 we have
(x – 2)2 + y2 = 4
b. Convert r = 4cos() into the rectangular form.
x
y
This is the circle centered at (2, 0)
with radius r = 2.

91. Example D. Convert 2x2 = 3x – 2y2 – 8 into a polar
equation.
Polar Coordinates & Graphs

92. 2x2 = 3x – 2y2 – 8
Example D. Convert 2x2 = 3x – 2y2 – 8 into a polar
equation.
Polar Coordinates & Graphs

93. 2x2 = 3x – 2y2 – 8
2x2 + 2y2 = 3x – 8
Example D. Convert 2x2 = 3x – 2y2 – 8 into a polar
equation.
Polar Coordinates & Graphs
grouping the square terms,

94. 2x2 = 3x – 2y2 – 8
2x2 + 2y2 = 3x – 8
2(x2 + y2) = 3x – 8
Example D. Convert 2x2 = 3x – 2y2 – 8 into a polar
equation.
Polar Coordinates & Graphs
grouping the square terms,

95. 2x2 = 3x – 2y2 – 8
2x2 + 2y2 = 3x – 8
2(x2 + y2) = 3x – 8
2r2 = 3rcos() – 8
Example D. Convert 2x2 = 3x – 2y2 – 8 into a polar
equation.
Polar Coordinates & Graphs
grouping the square terms,
converting into r and ,