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
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. 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)
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) The constant equation = c
requires that “the directional angle is c,
a fixed constant” and the distance r may
be of any value.
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. This equation describes
the line with polar angle c.
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.
x
y
The constant
equation = c
= c
25. Polar Coordinates & Graphs
For the graphs of other polar equations we need to plot
points accordingly.
26. 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.
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.
Example B. a. Graph r = 3cos().
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(). 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.
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. 3
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
–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.
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.
negative so the points are in the 4th quadrant.
Note the r’s are
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
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.
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.
In fact, they
trace over the
same points as goes from 0o to 90o
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.
3
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
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
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. 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
47. 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
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.
Multiply both sides by r so we
have r2 = 3r*cos(). 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().
In terms of x and y, it’s
x2 + y2 = 3x
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
x2 – 3x + y2 = 0
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
completing the square,
x2 – 3x + (3/2)2 + y2 = 0 + (3/2)2
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
(x – 3/2)2 + y2 = (3/2)2
3
53. 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.
54. 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
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
The points (r, ) and (r, –) are the
vertical mirror images of each other
across the x–axis.
x
(r, )
(r, –)
1
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. 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
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, –)
r = cos() = cos(–)
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, so r = ±D*cos()
are the horizontal circles.
x
(r, )
(r, –)
r = cos() = cos(–)
1
59. Polar Coordinates & Graphs
x
(r, )(–r, –)
y
The points (r, ) and (–r, –) are the
mirror images of each other across
the y–axis.
60. 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.
61. 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.
62. 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.
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().
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().
The graph of r = 1 – cos() is symmetric with respect
to the x–axis because cos() = 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(–). Therefore we
will plot from 0o to 180o and take its mirrored image
across the x–axis for the complete graph.
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. As goes
from 0o to 180o, cos() goes from 1 to –1, and the
expression 1 – cos() goes from 0 to 2.
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. The table is
shown below, readers may verify the approximate
values of r’s.
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
Reflecting across the x–axis, we have the cardioid.
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
x
2
Reflecting across the x–axis, we have the cardioid.
74. 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.
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
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
80. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
81. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
82. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
83. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
84. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
85. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
86. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
87. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
88. Polar Coordinates & Graphs
o
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown.
K
89. Polar Coordinates & Graphs
o
cases of polar equations of the form
r = a ± b*cos() and r = a ± b*sin().
K
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown. Cardioids are special
90. Polar Coordinates & Graphs
o
cases of polar equations of the form
r = a ± b*cos() and r = a ± b*sin().
K
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown. Cardioids are special
We summarize the graphs of these equations below.
91. Polar Coordinates & Graphs
o
cases of polar equations of the form
r = a ± b*cos() and r = a ± b*sin().
K
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown. Cardioids are special
We summarize the graphs of these equations below.
The graphs of r = a ± b*sin() are rotations of the
graphs of r = a ± b*cos() = a(1 ± k*cos())
where k is a constant.
92. Polar Coordinates & Graphs
o
cases of polar equations of the form
r = a ± b*cos() and r = a ± b*sin().
K
A cardioid is also the outline or the
envelope of a series of circles that
pass through some fixed point o
whose centers sit on another
circle K that contains o,
as shown. Cardioids are special
We summarize the graphs of these equations below.
The graphs of r = a ± b*sin() are rotations of the
graphs of r = a ± b*cos() = a(1 ± k*cos())
where k is a constant. Regarding the “a” as a scalar,
we reduce to examining the graphs of the polar
equations of the form r = 1 ± k*cos().
93. Polar Coordinates & Graphs
The graph of a polar equation of the form
r = 1 ± k*cos() depends on the value of |k|.
94. Polar Coordinates & Graphs
The graph of a polar equation of the form
r = 1 ± k*cos() depends on the value of |k|.
We may classify the graphs r = 1 – k*cos() where k is
a positive number into three types.
95. Polar Coordinates & Graphs
The graph of a polar equation of the form
r = 1 ± k*cos() depends on the value of |k|.
We may classify the graphs r = 1 – k*cos() where k is
a positive number into three types.
For k = 1, or r = 1 – cos(),
we get a cardioid.
r = 1 – cos()
k = 1
96. Polar Coordinates & Graphs
The graph of a polar equation of the form
r = 1 ± k*cos() depends on the value of |k|.
We may classify the graphs r = 1 – k*cos() where k is
a positive number into three types.
For k = 1, or r = 1 – cos(),
we get a cardioid.
For k < 1, e.g. r = 1 – ½ *cos(),
we have r > 0 for all ’s.
r = 1 – cos()
k = 1
97. Polar Coordinates & Graphs
The graph of a polar equation of the form
r = 1 ± k*cos() depends on the value of |k|.
We may classify the graphs r = 1 – k*cos() where k is
a positive number into three types.
For k = 1, or r = 1 – cos(),
we get a cardioid.
For k < 1, e.g. r = 1 – ½ *cos(),
we have r > 0 for all ’s.
This means the graph does not
pass through the origin.
r = 1 – cos()
k = 1
98. Polar Coordinates & Graphs
The graph of a polar equation of the form
r = 1 ± k*cos() depends on the value of |k|.
We may classify the graphs r = 1 – k*cos() where k is
a positive number into three types.
For k = 1, or r = 1 – cos(),
we get a cardioid.
r = 1 – k *cos()
0 < k < 1
r = 1 – cos()
k = 1For k < 1, e.g. r = 1 – ½ *cos(),
we have r > 0 for all ’s.
This means the graph does not
pass through the origin.
Instead, the cusp, i.e. the
pinched point at the origin of the
cardioid is pushed out as shown.
99. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
100. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0,
101. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown.
x
r = 1 – 2cos()
(–1, 0) (0, π/3)
102. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown. As goes from π/3 to
π/2, r increases from 0 to 1,
so the points traverse from (0, π/3 )
to (1, π/2).
x
r = 1 – 2cos()
(–1, 0) (0, π/3)
103. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown. As goes from π/3 to
π/2, r increases from 0 to 1,
x
r = 1 – 2cos()
(0, π/3)(–1, 0)
(1, π/2)
(3, π)
so the points traverse from (0, π/3 )
to (1, π/2).
104. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown. As goes from π/3 to
π/2, r increases from 0 to 1,
r increases from 1 to 3, and the points traverse from
(1, π/2) to (3, π).
x
r = 1 – 2cos()
(0, π/3)(–1, 0)
(1, π/2)
(3, π)
so the points traverse from (0, π/3 )
to (1, π/2). As from π/2 to π,
105. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown. As goes from π/3 to
π/2, r increases from 0 to 1,
r increases from 1 to 3, and the points traverse from
(1, π/2) to (3, π).
For 1 < k, r = 1 – k*cos()
has an inner loop.
x
r = 1 – 2cos()
(0, π/3)(–1, 0)
(1, π/2)
(3, π)
so the points traverse from (0, π/3 )
to (1, π/2). As from π/2 to π,
106. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown. As goes from π/3 to
π/2, r increases from 0 to 1,
r increases from 1 to 3, and the points traverse from
(1, π/2) to (3, π). Finally since cos() = cos(–), we
obtain the entire graph by taking its reflection across
the x–axis.
For 1 < k, r = 1 – k*cos()
has an inner loop.
x
r = 1 – 2cos()
(0, π/3)(–1, 0)
(1, π/2)
(3, π)
so the points traverse from (0, π/3 )
to (1, π/2). As from π/2 to π,
107. Polar Coordinates & Graphs
For k > 1, e.g. r = 1 – 2cos(),
we have r = –1 < 0 for = 0.
In fact, as goes from 0 to π/3,
r goes from –1 to 0, and the
corresponding points traverse
from (–1, 0) to (0, π/3) as
shown. As goes from π/3 to
π/2, r increases from 0 to 1,
so the points traverse from (0, π/3 )
to (1, π/2). As from π/2 to π,
r increases from 1 to 3, and the points traverse from
(1, π/2) to (3, π). Finally since cos() = cos(–), we
obtain the entire graph by taking its reflection across
the x–axis. Note that we have an inner loop if k > 1.
For 1 < k, r = 1 – k*cos()
has an inner loop.
x
r = 1 – 2cos()
(0, π/3)(–1, 0)
(1, π/2)
(3, π)
108. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos()
109. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos()
110. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos()
111. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos() r = 1 – 2cos()
112. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos() r = 1 – 2cos() r = 1 – 4cos()
113. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos() r = 1 – 2cos() r = 1 – 4cos()
Polar equations of the forms r = sin(n) or r = cos(n),
where n is a positive integer, form floral shape petals
that mathematicians call “roses”.
The Roses
114. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos() r = 1 – 2cos() r = 1 – 4cos()
Polar equations of the forms r = sin(n) or r = cos(n),
where n is a positive integer, form floral shape petals
that mathematicians call “roses”.
The Roses
115. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos() r = 1 – 2cos() r = 1 – 4cos()
Polar equations of the forms r = sin(n) or r = cos(n),
where n is a positive integer, form floral shape petals
that mathematicians call “roses”.
Recall that for n = 1, r = cos() consists of two
overlapping circles, i.e. the graph traverses the circle
twice as goes from 0 to 2π.
The Roses
116. Polar Coordinates & Graphs
Here is a sequence of graphs for r = 1 – kcos().
r = 1 – (1/4)cos() r = 1 – (1/2)cos() r = 1 – 1cos() r = 1 – 2cos() r = 1 – 4cos()
Polar equations of the forms r = sin(n) or r = cos(n),
where n is a positive integer, form floral shape petals
that mathematicians call “roses”.
Recall that for n = 1, r = cos() consists of two
overlapping circles, i.e. the graph traverses the circle
twice as goes from 0 to 2π. This is different from the
cases when n is even where the graph consists of 2n
petals.
The Roses
117. Polar Coordinates & Graphs
r = cos(1) r = cos(3)
If n is odd, the graph of r = cos(n) consists of
n petals as goes from 0 to π,
r = cos(7)r = cos(5)
118. Polar Coordinates & Graphs
r = cos(1) r = cos(3)
If n is odd, the graph of r = cos(n) consists of
n petals as goes from 0 to π, then the graph follows
the same path as goes from π to 2π.
r = cos(7)r = cos(5)
119. Polar Coordinates & Graphs
r = cos(1) r = cos(3)
If n is odd, the graph of r = cos(n) consists of
n petals as goes from 0 to π, then the graph follows
the same path as goes from π to 2π.
If n is even, the graph of r = cos(n) traces out
2n petals as goes from 0 to 2π.
r = cos(7)r = cos(5)
120. Polar Coordinates & Graphs
r = cos(1) r = cos(3)
If n is odd, the graph of r = cos(n) consists of
n petals as goes from 0 to π, then the graph follows
the same path as goes from π to 2π.
r = cos(2) r = cos(4) r = cos(6) r = cos(8)
If n is even, the graph of r = cos(n) traces out
2n petals as goes from 0 to 2π.
r = cos(7)r = cos(5)
121. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
122. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n even or odd.
Let’s look at the graph of r = | sin(3) |.
123. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
Let’s look at the graph of r = | sin(3) |.
For 0 ≤ < 2π, we have that 0 ≤ 3 < 6π.
124. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
Let’s look at the graph of r = | sin(3) |.
For 0 ≤ < 2π, we have that 0 ≤ 3 < 6π.
If 3 = 0, π, 2π, 3π, 4π, 5π, then r = sin(3) = 0,
or r = 0 when = 0, π/3, 2π/3, π, 4π/3, 5π/3
π/32π/3
The graph of
r = | sin(3) |
0
125. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
Let’s look at the graph of r = | sin(3) |.
For 0 ≤ < 2π, we have that 0 ≤ 3 < 6π.
If 3 = 0, π, 2π, 3π, 4π, 5π, then r = sin(3) = 0,
or r = 0 when = 0, π/3, 2π/3, π, 4π/3, 5π/3.
Similarly, r = 1 when = π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6.
π/32π/3
The graph of
r = | sin(3) |
0
126. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
As goes from 0 to π/6 to π/3, r goes
from 0 to 1 back to 0.
Let’s look at the graph of r = | sin(3) |.
For 0 ≤ < 2π, we have that 0 ≤ 3 < 6π.
If 3 = 0, π, 2π, 3π, 4π, 5π, then r = sin(3) = 0,
or r = 0 when = 0, π/3, 2π/3, π, 4π/3, 5π/3.
Similarly, r = 1 when = π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6.
π/32π/3
The graph of
r = | sin(3) |
0
127. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
(1, π/6)
π/3
The graph of
r = | sin(3) |
0
Let’s look at the graph of r = | sin(3) |.
For 0 ≤ < 2π, we have that 0 ≤ 3 < 6π.
If 3 = 0, π, 2π, 3π, 4π, 5π, then r = sin(3) = 0,
or r = 0 when = 0, π/3, 2π/3, π, 4π/3, 5π/3.
Similarly, r = 1 when = π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6.
2π/3As goes from 0 to π/6 to π/3, r goes
from 0 to 1 back to 0. So the graph
starts at the origin and makes a petal
(loop), to a tip of distance 1 from the
origin, back to the origin in a period
of π/3.
128. Polar Coordinates & Graphs
Following is a brief argument for the differences in the
graphs of r = sin(n) depending on n is even or odd.
(1, π/6)
As goes from 0 to π/6 to π/3, r goes
from 0 to 1 back to 0. So the graph
starts at the origin and makes a petal
(loop), to a tip of distance 1 from the
origin, back to the origin in a period
of π/3. Repeat this every π/3, we get
6 petals for r = | sin(3) |.
π/32π/3
The graph of
r = | sin(3) |
0
Let’s look at the graph of r = | sin(3) |.
For 0 ≤ < 2π, we have that 0 ≤ 3 < 6π.
If 3 = 0, π, 2π, 3π, 4π, 5π, then r = sin(3) = 0,
or r = 0 when = 0, π/3, 2π/3, π, 4π/3, 5π/3.
Similarly, r = 1 when = π/6, π/2, 5π/6, 7π/6, 3π/2, 11π/6.
129. Polar Coordinates & Graphs
Let’s now consider the signs of r = sin(3) as shown.
below.
0
π/32π/3
0
π/32π/3
++
+
–
– –
The signs of
r = sin(3)
The graph of
r = | sin(3) |
(1, π/6)
130. Polar Coordinates & Graphs
Let’s now consider the signs of r = sin(3) as shown.
below.
0
π/32π/3
0
π/32π/3
++
+
–
– –
The signs of
r = sin(3)
Note the difference in the signs of opposite segments.
The graph of
r = | sin(3) |
(1, π/6)
131. Polar Coordinates & Graphs
Let’s now consider the signs of r = sin(3) as shown.
below.
0
π/32π/3 π/32π/3
++
+
–
– –
The signs of
r = sin(3)
Note the difference in the signs of opposite segments.
Hence the “negative petals” flip across the origin in the
graph of r = sin(3) as shown.
The graph of
r = | sin(3) |
(1, π/6)
0
132. Polar Coordinates & Graphs
Let’s now consider the signs of r = sin(3) as shown.
below.
0
π/32π/3
0
π/32π/3
0
π/32π/3
++
+
–
– –
The signs of
r = sin(3)
Note the difference in the signs of opposite segments.
Hence the “negative petals” flip across the origin in the
graph of r = sin(3) of as shown.
The graph of
r = sin(3)
The graph of
r = | sin(3) |
– –
–
(1, π/6)
(1, π/6) =
(–1, 7π/6)
133. Polar Coordinates & Graphs
Let’s now consider the signs of r = sin(3) as shown.
below.
0
π/32π/3
0
π/32π/3
0
π/32π/3
++
+
–
– –
The signs of
r = sin(3)
Note the difference in the signs of opposite segments.
Hence the “negative petals” flip across the origin in the
graph of r = sin(3) of as shown. This is true in
general when n is odd, that the graph of r = sin(n)
consists of n petals because the “negative petals”
fold into the opposite positive ones
The graph of
r = sin(3)
The graph of
r = | sin(3) |
– –
–
(1, π/6)
(1, π/6) =
(–1, 7π/6)
134. Polar Coordinates & Graphs
Let’s now consider the signs of r = sin(3) as shown.
below.
0
π/32π/3
0
π/32π/3
0
π/32π/3
++
+
–
– –
The signs of
r = sin(3)
Note the difference in the signs of opposite segments.
Hence the “negative petals” flip across the origin in the
graph of r = sin(3) of as shown. This is true in
general when n is odd, that the graph of r = sin(n)
consists of n petals because the “negative petals”
fold into the opposite positive ones and the graph
traverses each petal twice as goes from 0 to 2π.
The graph of
r = sin(3)
The graph of
r = | sin(3) |
– –
–
(1, π/6)
(1, π/6) =
(–1, 7π/6)
135. Example C. Sketch the graph r = sin(5).
Polar Coordinates & Graphs
136. Example C. Sketch the graph r = sin(5).
The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
Polar Coordinates & Graphs
137. Example C. Sketch the graph r = sin(5).
The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
0
Polar Coordinates & Graphs
2π/5
π/5
r = sin(5)
138. (1, π/10) =
(–1, 11π/10)
Example C. Sketch the graph r = sin(5).
The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
0
Polar Coordinates & Graphs
2π/5
π/5
r = sin(5)
139. The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
Polar Coordinates & Graphs
(1, π/10) =
(–1, 11π/10)
0
2π/5
π/5
Example C. Sketch the graph r = sin(5).
+
–+–
+
+ +
– –
–
r = sin(5)
140. The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
Polar Coordinates & Graphs
(1, π/10) =
(–1, 11π/10)
0
2π/5
π/5
Example C. Sketch the graph r = sin(5).
+
–+–
+
+ +
– –
–
r = sin(5)
In general if n is odd, r = sin(n)
has n petals from 0 to 2π.
141. The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
Polar Coordinates & Graphs
π/4
+
–+
–
+
–
–
+
(1, π/10) =
(–1, 11π/10)
0
2π/5
π/5
Example C. Sketch the graph r = sin(5).
+
–+–
+
+ +
– –
–
r = sin(5)
In general if n is odd, r = sin(n)
has n petals from 0 to 2π.
If n is even, e.g. r = sin(4), then
its signs are distributed as shown,
i.e. two opposite wedges have
the same sign
r = sin(4)
142. The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
Polar Coordinates & Graphs
r = sin(4)
π/4
+
–+
–
+
–
–
+
If n is even, e.g. r = sin(4), then
its signs are distributed as shown,
i.e. two opposite wedges have
the same sign so the graph of
r = sin(4) retains all eight petals.
(1, π/10) =
(–1, 11π/10)
0
2π/5
π/5
Example C. Sketch the graph r = sin(5).
+
–+–
+
+ +
– –
–
r = sin(5)
In general if n is odd, r = sin(n)
has n petals from 0 to 2π.
143. The graph r = sin(5) consists
of 5 petals sitting evenly in 10
wedges each having a radial
angle of π/5.
Polar Coordinates & Graphs
r = sin(4)
π/4
+
–+
–
+
–
–
+
If n is even, e.g. r = sin(4), then
its signs are distributed as shown,
i.e. two opposite wedges have
the same sign so the graph of
r = sin(4) retains all eight petals.
In general if n is even, r = sin(n)
has 2n petals.
(1, π/10) =
(–1, 11π/10)
0
2π/5
π/5
Example C. Sketch the graph r = sin(5).
+
–+–
+
+ +
– –
–
r = sin(5)
In general if n is odd, r = sin(n)
has n petals from 0 to 2π.
145. Polar Equations
Spirals
A spiral is the graph of r = f() where f() is increasing
or decreasing.
Example D. a. Graph r = where ≥ 0.
146. Polar Equations
Spirals
A spiral is the graph of r = f() where f() is increasing
or decreasing.
Example D. a. Graph r = where ≥ 0.
The polar equation states that the
distance r is the same as .
147. Polar Equations
x
r =
Spirals
A spiral is the graph of r = f() where f() is increasing
or decreasing.
Example D. a. Graph r = where ≥ 0.
The polar equation states that the
distance r is the same as .
Starting at (0, 0), as increases,
r increases, so the points are circling
outward from the origin at a steady rate.
148. Polar Equations
A uniformly banded spiral such as this
one is called an Archimedean spiral.
x
The Archimedean Spiral
r =
Spirals
A spiral is the graph of r = f() where f() is increasing
or decreasing.
Example D. a. Graph r = where ≥ 0.
The polar equation states that the
distance r is the same as .
Starting at (0, 0), as increases,
r increases, so the points are circling
outward from the origin at a steady rate.
149. Polar Equations
A uniformly banded spiral such as this
one is called an Archimedean spiral.
x
The Archimedean Spiral
r =
Spirals
A spiral is the graph of r = f() where f() is increasing
or decreasing.
Example D. a. Graph r = where ≥ 0.
The polar equation states that the
distance r is the same as .
Starting at (0, 0), as increases,
r increases, so the points are circling
outward from the origin at a steady rate.
b. Convert the polar equation r =
into a rectangular equation by using
the cosine inverse to express .
151. Polar Equations
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
x
r =
x
152. Polar Equations
We have the equation that
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r =
x
153. Polar Equations
We have the equation that
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r = This rectangular equation only
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
x
154. Polar Equations
We have the equation that
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r = This rectangular equation only
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
cos–1(x/√x2 + y2) = √x2 + y2
x
x
The “Lost in Translation”
from the polar to the
rectangular equation
155. Polar Equations
We have the equation that
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r = This rectangular equation only
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
For other parts of the spirals,
we add nπ with n = 1,2,..
cos–1(x/√x2 + y2) = √x2 + y2
x
x
The “Lost in Translation”
from the polar to the
rectangular equation
156. Polar Equations
We have the equation that
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r = This rectangular equation only
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
cos–1(x/√x2 + y2) = √x2 + y2
x
x
The “Lost in Translation”
from the polar to the
rectangular equation
For other parts of the spirals,
we add nπ to with n = 1,2,.. to
obtain more distant segments, so
cos–1(x/√x2 + y2) + nπ = √x2 + y2.
157. Polar Equations
We have the equation that
We will use the cosine inverse
function to express in x&y, i.e.
= cos–1(x/r) = cos–1(x/√x2 + y2 ).
cos–1(x/√x2 + y2) = √x2 + y2 ( = r ) x
r = This rectangular equation only
gives the part of the spiral where
0 < √x2 + y2 ≤ π (why?)
cos–1(x/√x2 + y2) = √x2 + y2
x
x
The “Lost in Translation”
from the polar to the
rectangular equation
This shows the advantages of the
polar system in certain settings.
cos–1(x/√x2 + y2) + nπ = √x2 + y2.
For other parts of the spirals,
we add nπ to with n = 1,2,.. to
obtain more distant segments, so
159. Polar Coordinates & Graphs
The Log or Equiangular Spirals
The spirals r = aeb where a, b are constants are
called logarithmic spirals.
160. Polar Coordinates & Graphs
The Log or Equiangular Spirals
The spiral r = aeb where a, b are constants are called
logarithmic spirals.
The log–spirals, named after the log–form = β*ln(r)
of the equation r = eα, are also known as the
equiangular spirals.
161. Polar Coordinates & Graphs
The Log or Equiangular Spirals
The spiral r = aeb where a, b are constants are called
logarithmic spirals.
r = e0.15 r = e0.75
r = e0.35
The log–spirals, named after the log–form = β*ln(r)
of the equation r = eα, are also known as the
equiangular spirals.
Here are some examples of log–spirals.
162. Polar Coordinates & Graphs
The Log or Equiangular Spirals
The spiral r = aeb where a, b are constants are called
logarithmic spirals.
r = e0.15 r = e0.75
r = e0.35
The log–spirals, named after the log–form = β*ln(r)
of the equation r = eα, are also known as the
equiangular spirals.
Here are some examples of log–spirals.
Log–spirals are also called equiangular spirals
because of its geometric characteristic.
163. Polar Coordinates & Graphs
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
164. Polar Coordinates & Graphs
k
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
Equiangular spirals
165. Polar Coordinates & Graphs
k
k
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
Equiangular spirals
166. Polar Coordinates & Graphs
k
k
q
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
Equiangular spirals
167. Polar Coordinates & Graphs
q
k
k
q
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
Equiangular spirals
168. Polar Coordinates & Graphs
q
k
k
q
Equiangular spirals
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
If the curve spirals outward, the tangential angle must
be more than π/2
169. Polar Coordinates & Graphs
q
k
k
q
Equiangular spirals
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
If the curve spirals outward, the tangential angle must
be more than π/2. If the tangential angle is π/2,
the spiral contracts into a circle.
170. Polar Coordinates & Graphs
q
k
k
q
Equiangular spirals
The geometric significance of an equiangular spiral
is that the angle between the tangent line and
the radial line at any point on the spiral is a fixed
constant.
If the curve spirals outward, the tangential angle must
be more than π/2. If the tangential angle is π/2,
the spiral contracts into a circle. If the tangential angle
is less than π/2, then it’s spiraling inward toward (0,0).
172. Polar Coordinates & Graphs
Equiangular spirals occur in nature frequently.
In many biological growth processes, the new growth
is extruded at a fixed angle from the existing mass,
e.g. the generation of new sea shell along the edge of
the old shell.
http://en.wikipedia.
org/wiki/Logarithmi
c_spiral
173. Polar Coordinates & Graphs
Equiangular spirals occur in nature frequently.
In many biological growth processes,
the new growth is extruded at a fixed angle from the
existing structure, e.g. the growth of new sea shell
along the edge of the old shell, or the growth of plants.
Over time the equiangular–spiral growth lines emerge.
http://en.wikipedia.
org/wiki/Logarithmi
c_spiral
175. Polar Coordinates & Graphs
Let r1 = cos() = f() and
r2 = cos( – π/4) = g() so that
r2 = g( + π/4) = f() = r1.
Rotations of Polar Graphs
176. Polar Coordinates & Graphs
Let r1 = cos() = f() and
r2 = cos( – π/4) = g() so that
r2 = g( + π/4) = f() = r1. Therefore,
the point (r2, + π/4) on the graph of g
is the point (r1, ) on the graph of f
rotated by the angle π/4 as shown.
x
y
r = cos()
Rotations of Polar Graphs
177. Polar Coordinates & Graphs
Let r1 = cos() = f() and
r2 = cos( – π/4) = g() so that
r2 = g( + π/4) = f() = r1. Therefore,
the point (r2, + π/4) on the graph of g
is the point (r1, ) on the graph of f
rotated by the angle π/4 as shown.
x
y
r = cos()
r = cos( – π/4)
π/4
Rotations of Polar Graphs
178. Polar Coordinates & Graphs
Let r1 = cos() = f() and
r2 = cos( – π/4) = g() so that
r2 = g( + π/4) = f() = r1. Therefore,
the point (r2, + π/4) on the graph of g
is the point (r1, ) on the graph of f
rotated by the angle π/4 as shown.
x
y
r = cos()
r = cos( – π/4)
π/4
Rotations of Polar Graphs
In general, given r1 = f(), let r2 = g() = f( – α)
so that g( + α) = f(), then the point (r2, + α)
is the point (r1, ) on the graph of f, rotated by α.
179. Polar Coordinates & Graphs
Let r1 = cos() = f() and
r2 = cos( – π/4) = g() so that
r2 = g( + π/4) = f() = r1. Therefore,
the point (r2, + π/4) on the graph of g
is the point (r1, ) on the graph of f
rotated by the angle π/4 as shown.
x
y
r = cos()
r = cos( – π/4)
π/4
Rotations of Polar Graphs
In general, given r1 = f(), let r2 = g() = f( – α)
so that g( + α) = f(), then the point (r2, + α)
is the point (r1, ) on the graph of f, rotated by α.
If α is positive, then the graph of r2 = f( – α) is the
counter clockwise rotation of r1 = f() by the angle α,
180. Polar Coordinates & Graphs
Let r1 = cos() = f() and
r2 = cos( – π/4) = g() so that
r2 = g( + π/4) = f() = r1. Therefore,
the point (r2, + π/4) on the graph of g
is the point (r1, ) on the graph of f
rotated by the angle π/4 as shown.
x
y
r = cos()
r = cos( – π/4)
π/4
Rotations of Polar Graphs
In general, given r1 = f(), let r2 = g() = f( – α)
so that g( + α) = f(), then the point (r2, + α)
is the point (r1, ) on the graph of f, rotated by α.
If α is positive, then the graph of r2 = f( – α) is the
counter clockwise rotation of r1 = f() by the angle α,
and the graph of r2 = f( + α) is the clockwise rotation
of r1 = f() by the angle α.
182. Polar Coordinates & Graphs
The point (2r, ) is the radial extension of
the point (r, ) as shown.
Radial Extensions/Contractions
of Polar Graphs
O
(r, )pr
183. Polar Coordinates & Graphs
The point (2r, ) is the radial extension of
the point (r, ) as shown.
Radial Extensions/Contractions
of Polar Graphs
O
(r, )pr
(2r, )p
184. Polar Coordinates & Graphs
The point (2r, ) is the radial extension of
the point (r, ) as shown.
Radial Extensions/Contractions
of Polar Graphs
O
(r, )pr
In general, the graph of r = kf()
when k is a constant,
is the radial stretch/compression
of the graph r = f().
(2r, )p
185. Polar Coordinates & Graphs
The point (2r, ) is the radial extension of
the point (r, ) as shown.
Radial Extensions/Contractions
of Polar Graphs
O
(r, )pr
x
y
r = cos( – π/4)
r = 2cos( – π/4)
radial stretch/compression k >0
In general, the graph of r = kf()
when k is a constant,
is the radial stretch/compression
of the graph r = f().
The graphs of
r1 = cos ( – π/4),
(2r, )p
186. Polar Coordinates & Graphs
The point (2r, ) is the radial extension of
the point (r, ) as shown.
Radial Extensions/Contractions
of Polar Graphs
O
(r, )pr
x
y
r = cos( – π/4)
r = 2cos( – π/4)
radial stretch/compression k >0
In general, the graph of r = kf()
when k is a constant,
is the radial stretch/compression
of the graph r = f().
The graphs of
r1 = cos ( – π/4),
r2 = 2cos ( – π/4)
(2r, )p
187. Polar Coordinates & Graphs
The point (2r, ) is the radial extension of
the point (r, ) as shown.
Radial Extensions/Contractions
of Polar Graphs
O
(r, )pr
x
y
r = cos( – π/4)
r = 2cos( – π/4)
radial stretch/compression k >0
r = ½ cos( – π/4)
In general, the graph of r = kf()
when k is a constant,
is the radial stretch/compression
of the graph r = f().
The graphs of
r1 = cos ( – π/4),
r2 = 2cos ( – π/4) and
r3 = ½ * cos ( – π/4)
are shown here as examples.
(2r, )p
188. Polar Coordinates & Graphs
If k < 0, then the graph of r = kf() is the
diagonal reflection across the origin
of r = |k| f().
189. Polar Coordinates & Graphs
If k < 0, then the graph of r = kf() is the
diagonal reflection across the origin
of r = |k| f(). Here are the graphs
of r = –2cos ( – π/4)
and r = 2cos ( – π/4).
x
r = 2cos( – π/4)
r = –2cos( – π/4)
r = cos( – π/4)
Diagonal
Reflection
with k < 0