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Polar Coordinates
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
Objective:
*  Introduction to polar coordinates
   and conversion between 
   polar and rectangular coordinates
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Polar Coordinates
The loca= tion of a point P in the plane may be given by two numbers:=
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Polar Coordinates
The loca= tion of a point P in the plane may be given by two numbers:=
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Polar Coordinates
The loca= tion of a point P in the plane may be given by two numbers:=
r =3D distance of = the location to the origin
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Polar Coordinates
The loca= tion of a point P in the plane may be given by two numbers:=
r =3D distance of = the location to the origin
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
The loca= tion of a point P in the plane may be given by two numbers:=
r =3D distance of = the location to the origin
q =3D directional angle measured against the +x-axis.
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Polar Coordinates
The loca= tion of a point P in the plane may be given by two numbers:=
r =3D distance of = the location to the origin
q =3D directional angle measured against the +x-axis.
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Content-Type: image/gif R0lGODlhFQAWAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAU ABUAgAAAAJnMAAI/hI8Zka27nITQMAmozbrFo3XgZZGlaXIe+XEt8kXUO55tCp7KxNAZr4LhSsGa I4YaFis+mLDpxAClR6UUaigAADs= ------=_NextPart_01C97CE7.55302D00 Content-Location: file:///C:/09764A33/41PolarCoordinateandEquations_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinates
<= span style=3D'font-size:88%'>The location of a point P in the <= span style=3D'font-size:88%'>plane may be given by two <= span style=3D'font-size:88%'>numbers:
r =3D distance of the location to
the origin
q =3D directional angle measured
against the +x-axis.
<= span style=3D'font-size:88%'>The ordered pair (r, <= /span>q) is the polar coordinate of the point P.
(r, q)
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file:///C:/09764A33/41PolarCoordinateandEquations_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
 
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
------=_NextPart_01C97CE7.55302D00 Content-Location: file:///C:/09764A33/41PolarCoordinateandEquations_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
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Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
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Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (<= /span>-4, 135o)P 
Negative r means to go in the opposite direction.
Negative r means to go in the opposite direction. 
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Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)R for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (<= /span>-4, 135o)P 
Negative r means to go in the opposite direction.
Negative r means to go in the opposite direction.  <= /div>
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinates
If needed, we write (a, b)P for
polar coordinate ordered pair,
and (a, = b)C for rectangular <= span style=3D'font-size:88%'>coordinate ordered pair.
E= xample: Plot the following
c= oordinate pairs:
A= =3D (4, 60o)P ,   B= =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (<= /span>-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.
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
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file:///C:/09764A33/41PolarCoordinateandEquations_files/slide0022.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinates
Conversion Rule:
Let (x, y)R and (r, q)P be
the rectangular and <= span style=3D'mso-hansi-font-family:"Times New Roman";color:black;mso-color-ind= ex: 3'>polar coordinates of the same point P, then:
&= #13;
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Polar Coordinates
Conversion Rule:
Let (x, y)R and (r, q)P be
the rectangular and <= span style=3D'mso-hansi-font-family:"Times New Roman";color:black;mso-color-ind= ex: 3'>polar coordinates of the same point P, then:
x =3D r*cos(q)
y =3D r*sin(q)
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
Conversion Rule:
Let (x, y)R and (r, q)P be
= the rectangular and polar coordinates of the = same point P, then= :
x =3D r*cos(q)
y =3D r*sin(q)
r2 =3D x2 + y2, 
&= #13;
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Polar Coordinates
Conversion Rule:
Let (x, y)R and (r, q)P be
= the rectangular and polar coordinates of the = same point P, then= :
x =3D r*cos(q)
y =3D r*sin(q)
r2 =3D x2 + y2, tan(q) =3D y/x
&= #13;
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D (4*cos(60), 4*sin(60))
=         
Ó=  <= /span>= Fra= nk Ma 2= 006
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D (4*cos(60), 4*sin(60))
=          =3D  (2, 2Ö3)
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Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D (4*cos(60), 4*sin(60))
=          =3D  (2, 2Ö3)
= For B =3D (8, 0o)P ,
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D (4*cos(60), 4*sin(60))
=          =3D  (2, 2Ö3)
= For B =3D (8, 0o)P , (x, y) = =3D (8, 0)
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D (4*cos(60), 4*sin(60))
=          =3D  (2, 2Ö3)
= For B =3D (8, 0o)P , (x, y) = =3D (8, 0)
For C and D,  =
(x, y) =3D (4cos(-45o), 4sin(-45= o))
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Polar Coordinates
= Example: Convert the following
= into rectangular coordinate pairs:
= A =3D (4, 60o)P ,   B =3D (8, 0o)P
C =3D (4, -45o)P ,  D =3D (-= 4, 135o)P 
= For A =3D (4, 60o)P ,  
= (x, y) =3D (4*cos(60), 4*sin(60))
=          =3D  (2, 2Ö3)
= For B =3D (8, 0o)P , (x, y) = =3D (8, 0)
For C and D, 
(x, y) =3D (4cos(-45o), 4sin(-45= o)) =3D (-4cos(135), -4sin(135))=3D(2= Ö2, -2Ö2)
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Polar Coordinates
= Example: Convert the following
= polar coordinate pairs:
= A =3D (-4, 6)R ,   B =3D (-8, 0)R,  C =3D (= -3, -2)R 
Ó=  <= /span>= Fra= nk Ma 2= 006
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Coordinate
Polar Coordinates
= Example: Convert the following
= polar coordinate pairs:
= A =3D (-4, 6)R ,   B =3D (-8, 0)R,  C =3D (= -3, -2)R 
For A, r =3D Ö16+36 =3D <= /span>Ö52