41 polar coordinate and equations
TRANSCRIPT
Polar Coordinates
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Polar Coordinates
Objective:
* Introduction to polar coordinates
and conversion between
polar and rectangular coordinates
Polar CoordinatesThe location of a point P in the
plane may be given by two
numbers:
Polar CoordinatesThe location of a point P in the
plane may be given by two
numbers:
Polar CoordinatesThe location of a point P in the
plane may be given by two
numbers:
r = distance of the location to
the origin
Polar CoordinatesThe location of a point P in the
plane may be given by two
numbers:
r = distance of the location to
the origin
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Polar CoordinatesThe location of a point P in the
plane may be given by two
numbers:
r = distance of the location to
the origin
= directional angle measured
against the +x-axis.
Polar CoordinatesThe location of a point P in the
plane may be given by two
numbers:
r = distance of the location to
the origin
= directional angle measured
against the +x-axis.
Polar CoordinatesThe location of a point P in the plane may be given by two numbers:
r = distance of the location to
the origin
= directional angle measured
against the +x-axis.
The ordered pair (r, ) is the polar coordinate of the point P.
(r, )
Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
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Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
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Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Negative r means to go in the opposite direction.Negative r means to go in the opposite direction.
Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)R for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Negative r means to go in the opposite direction.Negative r means to go in the opposite direction.
Polar CoordinatesIf needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, b)C for rectangular
coordinate ordered pair.
Example: Plot the following
coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Negative r means to go in the opposite direction.Negative r means to go in the opposite direction.
There are infinite many ordered pairs representing each position.
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Polar Coordinates
Conversion Rule:
Polar Coordinates
Conversion Rule:
Let (x, y)R and (r, )P be
the rectangular and
polar coordinates of the
same point P, then:
Polar Coordinates
Conversion Rule:
Let (x, y)R and (r, )P be
the rectangular and
polar coordinates of the
same point P, then:
x = r*cos()
y = r*sin()
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Polar CoordinatesConversion Rule:
Let (x, y)R and (r, )P be
the rectangular and polar coordinates of the same point P, then:
x = r*cos()
y = r*sin()
r2 = x2 + y2,
Polar CoordinatesConversion Rule:
Let (x, y)R and (r, )P be
the rectangular and polar coordinates of the same point P, then:
x = r*cos()
y = r*sin()
r2 = x2 + y2, tan() = y/x
Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) =
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Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) = (4*cos(60), 4*sin(60))
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Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) = (4*cos(60), 4*sin(60))
= (2, 23)
Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) = (4*cos(60), 4*sin(60))
= (2, 23)
For B = (8, 0o)P ,
Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) = (4*cos(60), 4*sin(60))
= (2, 23)
For B = (8, 0o)P , (x, y) = (8, 0)
Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) = (4*cos(60), 4*sin(60))
= (2, 23)
For B = (8, 0o)P , (x, y) = (8, 0)
For C and D,
(x, y) = (4cos(-45o), 4sin(-45o))
Polar CoordinatesExample: Convert the following
into rectangular coordinate pairs:
A = (4, 60o)P , B = (8, 0o)P
C = (4, -45o)P , D = (-4, 135o)P
For A = (4, 60o)P ,
(x, y) = (4*cos(60), 4*sin(60))
= (2, 23)
For B = (8, 0o)P , (x, y) = (8, 0)
For C and D,
(x, y) = (4cos(-45o), 4sin(-45o)) = (-4cos(135), -4sin(135))=(22, -22)
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
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Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
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Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
For B, r = 8, = 180o
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
For B, r = 8, = 180o
Hence B = (8, 180o)P
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
For B, r = 8, = 180o
Hence B = (8, 180o)P
For C, r = 9 + 4 = 13,
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
For B, r = 8, = 180o
Hence B = (8, 180o)P
A
For C, r = 9 + 4 = 13, tan(A) = 2/3, A = tan-1(A) 33.7o
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Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
For B, r = 8, = 180o
Hence B = (8, 180o)P
A
For C, r = 9 + 4 = 13, tan(A) = 2/3, A = tan-1(A) 33.7o
180o+A 213.7o,
Polar CoordinatesExample: Convert the following
polar coordinate pairs:
A = (-4, 6)R , B = (-8, 0)R, C = (-3, -2)R
For A, r = 16+36 = 52
= arccos(-4/52) 123o
Hence A (52, 123o)
For B, r = 8, = 180o
Hence B = (8, 180o)P
A
For C, r = 9 + 4 = 13, tan(A) = 2/3, A = tan-1(A) 33.7o
180o+A 213.7o, hence C = (13, 213.7o)
Polar Equations
Polar Equations
Polars equations are equations in the
variables r and . Many curves may be
described easier using relations in r and
rather than relations between x and y.
Polar Equations
Polars equations are equations in the
variables r and . Many curves may be
described easier using relations in r and
rather than relations between x and y.
Polar Equations
Polars equations are equations in the
variables r and . Many curves may be
described easier using relations in r and
rather than relations between x and y.
Polar Equations
The Constant Equations r = c & =c
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Polar Equations
I. The equations r = c,
The Constant Equations r = c & =c
Polar Equations
I. The equation r = c,
distance from the point to the origin = c, and any number.
The Constant Equations r = c & =c
Polar Equations
The Constant Equations r = c & =c
I. The equation r = c,
distance from the point to the origin = c, and any number.
This equation describes the circle of radius c, centered at (0,0).
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Polar Equations
The Constant Equations r = c & =c
I. The equation r = c,
distance from the point to the origin = c, and any number.
This equation describes the circle of radius c, centered at (0,0).
Polar Equations
II. The equation = c,
The Constant Equations r = c & =c
Polar Equations
II. The equation = c,
Directional angle to the point = c, and r any number.
The Constant Equations r = c & =c
Polar Equations
II. The equation = c,
Directional angle to the point = c, and r any number.
This equation describes the line making the angle c to x-axis.
The Constant Equations r = c & =c
Polar Equations
II. The equation = c,
Directional angle to the point = c, and r any number.
This equation describes the line making the angle c to x-axis.
r>0
r<0
The Constant Equations r = c & =c
=C
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Polar Equations
r = ±c*cos() & r = ±c*sin()
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
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Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
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Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
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Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
F
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Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
F
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0
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Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
F
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n
k
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a
2
0
0
6
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
The
equations
r = ±c*cos()
r = ±c*sin()
are circles.
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
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Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
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Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2π
F
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2
0
0
6
Polar Equations
r = ±c*cos() & r = ±c*sin()
Example: Graph r = -sin()
r
0 0
-½ π/6
-2/2 π/4
-3/2 π/3
-1 π/2
-3/2 2π/3
-2/2 3π/4
-½ 5π/6
0 π
r
0 π
½ 7π/6
2/2 5π/4
3/2 4π/3
1 3π/2
3/2 5π/3
2/2 7π/4
½ 11π/6
0 2πNote the graph swept traced out two circles as goes from 0 to 2π.
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
F
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2
0
0
6
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
F
r
a
n
k
M
a
2
0
0
6
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
F
r
a
n
k
M
a
2
0
0
6
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
F
r
a
n
k
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a
2
0
0
6
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
F
r
a
n
k
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a
2
0
0
6
Polar Equations
r = c(1 ± cos()) & r =c(1 ± sin())
Example: Graph r = 1 – sin()
r
1 π
3/2 7π/6
1+2/2 5π/4
1+3/2 4π/3
2 3π/2
1+3/2 5π/3
1+2/2 7π/4
3/2 11π/6
1 2π
r
1 0
½ π/6
1-2/2 π/4
1-3/2 π/3
0 π/2
1-3/2 2π/3
1-2/2 3π/4
½ 5π/6
1 π
Polar Equationsr = cos(n) & r = c*sin(n)
Polar Equationsr = c*cos(n) & r = c*sin(n)
The following steps help us to graph
polar equations, especially equations
made up with sine and cosine of :
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2
0
0
6
Polar Equationsr = cos(n) & r = c*sin(n)
The following steps help us to graph
polar equations, especially equations
made up with sine and cosine of :
1. Find 0o < < 360o where r=0
Polar Equationsr = cos(n) & r = c*sin(n)
The following steps help us to graph
polar equations, especially equations
made up with sine and cosine of :
1. Find 0o < < 360o where r=0
2. Find between 0 and 360o where
|r| is greatest.
Polar Equationsr = cos(n) & r = c*sin(n)
The following steps help us to graph
polar equations, especially equations
made up with sine and cosine of :
1. Find 0o < < 360o where r=0
2. Find between 0 and 360o where
|r| is greatest.
3. Trace the curves using 1 and 2.
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0
0
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Polar EquationsExample: r = sin(2)
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2),
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
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2
0
0
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Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Find r = 1 = sin(2),
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Find r = 1 = sin(2), 2 = 90, 450, or = 45, 225.
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2
0
0
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Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Find r = 1 = sin(2), 2 = 90, 450, or = 45, 225.
Find r = -1,
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Find r = 1 = sin(2), 2 = 90, 450, or = 45, 225.
Find r = -1, 2 = 270, 630, = 135, 315
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Find r = 1 = sin(2), 2 = 90, 450, or = 45, 225.
Find r = -1, 2 = 270, 630, = 135, 315
0
90
180
270
Draw the directions that r = 0.
1
Polar EquationsExample: r = sin(2)
Find r = 0 = sin(2), 0 < < 360 0 < 2 < 720.
Therefore 2 = 0, 180, 360, 540, or = 0, 90, 180, 270
Find r = 1 = sin(2), 2 = 90, 450, or = 45, 225.
Find r = -1, 2 = 270, 630, = 135, 315
Draw the directions that r = 0.
0
90
180
270
1
45
225
135
315
Draw the directions that r = ±1.
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2
0
0
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Polar EquationsExample: r = sin(2)
0
90
180
270
1
45
225
135
315
Investigate the graph in each sector from r = 0 to r = 0 :
F
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2
0
0
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Polar EquationsExample: r = sin(2)
0
90
180
270
1
45
225
135
315
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
F
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n
k
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a
2
0
0
6
Polar EquationsExample: r = sin(2)
0
90
180
270
1
45
225
135
315
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
F
r
a
n
k
M
a
2
0
0
6
Polar EquationsExample: r = sin(2)
0
90
180
270
1
45
225
135
315
90 < <180 180 < 2 < 360, sin(2) goes from 0 to -1 to 0.
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
Polar EquationsExample: r = sin(2)
0
90
180
270
1
45
225
135
315
90 < <180 180 < 2 < 360, sin(2) goes from 0 to -1 to 0.
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
Polar EquationsExample: r = sin(2)
0
90
180
270
1
45
225
135
315
90 < <180 180 < 2 < 360, sin(2) goes from 0 to -1 to 0.
The similar observation
about the other two sectors
gives us the complete graph.
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
F
r
a
n
k
M
a
2
0
0
6
Polar EquationsExample: r = sin(2)
90 < <180 180 < 2 < 360, sin(2) goes from 0 to -1 to 0.
The similar observation
about the other two sectors
gives us the complete graph.
0
90
180
270
1
45
225
135
315
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
Polar EquationsExample: r = sin(2)
90 < <180 180 < 2 < 360, sin(2) goes from 0 to -1 to 0.
The similar observation
about the other two sectors
gives us the complete graph.
0
90
180
270
1
45
225
135
315
This is known as the four-
pedal-rose curve.
Investigate the graph in each sector from r = 0 to r = 0 :
0 < < 90 0 < 2 < 180, sin(2) goes from 0 to 1 back to 0.
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0
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Equation Conversion
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
Equation ConversionConversion Rule:
To convert equations between the polar and rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2, r = x2 + y2
tan() = y/x
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2
0
0
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Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 into
rectangular equation in x&y.
F
r
a
n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 into
rectangular equation in x&y.
r = 3 square both sides
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 into
rectangular equation in x&y.
r = 3 square both sides
r2 = 9
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 into
rectangular equation in x&y.
r = 3 square both sides
r2 = 9 replace into x&y
F
r
a
n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 into
rectangular equation in x&y.
r = 3 square both sides
r2 = 9 replace into x&y
x2 + y2 = 9
F
r
a
n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
r = 3 – 3cos()
F
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n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
r = 3 – 3cos(), multiply by r
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
r = 3 – 3cos(), multiply by r
r2 = 3r – 3*r*cos()
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
r = 3 – 3cos(), multiply by r
r2 = 3r – 3*r*cos() in x&y
F
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a
n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
r = 3 – 3cos(), multiply by r
r2 = 3r – 3*r*cos() in x&y
x2 + y2 = 3x2 + y2 – 3x
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the polar
equation r = 3 – 3cos() into
rectangular equation in x&y.
r = 3 – 3cos(), multiply by r
r2 = 3r – 3*r*cos() in x&y
x2 + y2 = 3x2 + y2 – 3x
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the
rectangular equation
2x2 = 3x – 2y2 – 8 into polar
equation.
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the
rectangular equation
2x2 = 3x – 2y2 – 8 into polar
equation.
2x2 = 3x – 2y2 – 8
F
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a
n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the
rectangular equation
2x2 = 3x – 2y2 – 8 into polar
equation.
2x2 = 3x – 2y2 – 8
2x2 + 2y2 = 3x – 8
F
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a
n
k
M
a
2
0
0
6
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the
rectangular equation
2x2 = 3x – 2y2 – 8 into polar
equation.
2x2 = 3x – 2y2 – 8
2x2 + 2y2 = 3x – 8
2(x2 + y2) = 3x – 8
Equation Conversion
Conversion Rule:
To convert equations
between the polar and
rectangular form:
x = r*cos()
y = r*sin()
r2 = x2 + y2
tan() = y/x
Example: Convert the
rectangular equation
2x2 = 3x – 2y2 – 8 into polar
equation.
2x2 = 3x – 2y2 – 8
2x2 + 2y2 = 3x – 8
2(x2 + y2) = 3x – 8
2r2 = 3rcos() – 8