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&Oac= ute;=  F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Ó= ; 2006, Frank = Ma
Polar Coordinate= s
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Objective= :
*  Introduction to polar coordinates=
   and conversion between 
   polar and rectangular coord= inates
Polar Coordinate= s
------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0066.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
The locat= ion of a point Q =3D(a, b)
in the pl= ane may be given by the following two numbers:
x
(a, b)
Q=3D
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Polar Coordinate= s
The locat= ion of a point Q =3D(a, b)
in the pl= ane may be given by the following two numbers:
r =3D distance of the location to
the origi= n
x
r
(a, b)
Q=3D
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Polar Coordinate= s
The locat= ion of a point Q =3D(a, b)
in the pl= ane may be given by the following two numbers:
r =3D distance of the location to
the origi= n
q =3D directional angle measured
against t= he +x-axis.
x
r
(a, b)
Q=3D
q
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=3D (r, = q)p
Polar Coordinate= s
The locat= ion of a point Q =3D(a, b)
in the pl= ane may be given by the following two numbers:
r =3D distance of the location to
the origi= n
q =3D directional angle measured
against t= he +x-axis.
x
r
(a, b)
Q=3D
q
The ordered pair (r, = q) is called the polar coordinate of Q. If needed, we write (r, q)for polar coordinates.
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=3D (r, = q)p
Polar Coordinate= s
The locat= ion of a point Q =3D(a, b)
in the pl= ane may be given by the following two numbers:
r =3D distance of the location to
the origi= n
q =3D directional angle measured
against t= he +x-axis.
x
r
Note that r and q are the same ones we used in the = cis-form = (polar form) for complex numbers.
(a, b)
Q=3D
q
The ordered pair (r, = q) is called the polar coordinate of Q. If needed, we write (r, q)for polar coordinates.
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Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P 
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Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
A =3D (8, 60o)P
8
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P 
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Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
A =3D (8, 60o)P
60o
8
B =3D (4, 0o)P
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P 
------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0054_image021.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwA5 AAUAgAAAAAAAAAIbhI+py70RDpy02ouzhmD7D17OSJbKZKZqEx0FADs= ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0054_image022.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgCoAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAG AKMAgAAAAAAAAAJSBBJmqMjnoEtQ1UZXvtq+jnnbx5XkOabiGrbga6puzMJoTc+3Ltt97sPtgjzh bwhMIpfHpvFZjBKnSqe0CqUys9jr1mvVhrti7rcMNo/P5Pa3AAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0054_image023.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOQBjAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 AGAAgQAAAAAAALvg4/8AAALOhI+pyxv/mpxUBYFxqLyjm2WbR05gKIzlmpyoysYAOISwvF5DLeLy vuv5WMAdZIgDIpeDpfMJjUqnu+mqavVgs50tl+L9TsLisvmMTpOaaga5/Yaz24i42p6e0/f8vj+p t4eHNmhWaBhId/jH2Oj4eLYoJslFWZko14gJydnp6WSZFUq1eVdKeIr6ucraeqWZisg4KkVbG2vo qrvLW4c7+Qv8ZxtF/GTcm6xcORv8hczkfJl5UFSUYB2YrZ2NrWSw7X3tay0+Xl0u+M3nVQAAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0054_image024.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUwADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABQ AAIAgAAAAAAAAAILjI+py+0Po5y0rgIAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0054_image025.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAK AAkAgQAAAAAAALvg4wECAwIUBIJpJ+3B3IuyofoCtnR2AS0JshQAOx== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0055.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
A =3D (8, 60o)P
60o
8
B =3D (4, 0o)P
-45o
4
C =3D (4, -45o)P
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P 
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Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
A =3D (8, 60o)P
60o
8
B =3D (4, 0o)P
-45o
4
C =3D (4, -45o)P
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P
Negative r means we go in the opposite direction of q.
135o
D =3D (-4, 135o)P
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Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
A =3D (8, 60o)P
60o
8
B =3D (4, 0o)P
-45o
4
C =3D (4, -45o)P
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P
Negative r means we go in the opposite direction of q.
135o
Hence,  D =3D (-4, 135o)P is the = same as C =3D (4, -45o)P. 
D =3D (-4, 135o)P
------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0057_image037.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwA5 AAUAgAAAAAAAAAIbhI+py70RDpy02ouzhmD7D17OSJbKZKZqEx0FADs= ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0057_image038.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgCoAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAG AKMAgAAAAAAAAAJSBBJmqMjnoEtQ1UZXvtq+jnnbx5XkOabiGrbga6puzMJoTc+3Ltt97sPtgjzh bwhMIpfHpvFZjBKnSqe0CqUys9jr1mvVhrti7rcMNo/P5Pa3AAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0057_image039.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOQBjAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 AGAAgQAAAAAAALvg4/8AAALOhI+pyxv/mpxUBYFxqLyjm2WbR05gKIzlmpyoysYAOISwvF5DLeLy vuv5WMAdZIgDIpeDpfMJjUqnu+mqavVgs50tl+L9TsLisvmMTpOaaga5/Yaz24i42p6e0/f8vj+p t4eHNmhWaBhId/jH2Oj4eLYoJslFWZko14gJydnp6WSZFUq1eVdKeIr6ucraeqWZisg4KkVbG2vo qrvLW4c7+Qv8ZxtF/GTcm6xcORv8hczkfJl5UFSUYB2YrZ2NrWSw7X3tay0+Xl0u+M3nVQAAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0057_image040.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUwADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABQ AAIAgAAAAAAAAAILjI+py+0Po5y0rgIAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0057_image041.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAAtAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA2 ACgAgQAAAAAAALvg4/8AAAJzhI+pyw0Bo5u0PoFzsLyHDG7dOH2gNpCqYp5CMKTr3J5bLM9kjYkA rtvZErHgLuIzKpfMpnNSfOqi0hWwqrpiR9St9wsOf7tiB7nM0KLT5zUi547LpfD5oT3Hy9X2n95d 1ydIEbj3h4ZTuJaouMjXd3hQAAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0057_image042.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUQBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAQABQBM AEwAgAAAAP8AAAKdTICpy+0P4zNG2oszolX7n3HgSEJiiaIH0KXuC8fyPLf03dj4vq38DwwKf7ph rGh8IZPMppO5fIKiUg+1is1qNddtpOud+MLkshlsVqDTPbb77Vyz5Wk6/I6fjvMLe3nPFyg44kdW GAY4qLj4lRh46OXIyCiZB7l1qVU5Obh5l5kFiuXJyUf6JlqVKnVa+sm5+hQb5+raOndbl3tWAAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Ó F= = r= a= = n= k<= span lang=3DEN-US style=3D'font-size:56%;color:red;mso-fareast-language:ZH-CN'>=  M= = a = 2= = 0= 0= 6
Polar Coordinate= s
A =3D (8, 60o)P
60o
8
B =3D (4, 0o)P
-45o
4
C =3D (4, -45o)P
Example: Plot the following
coordinate pairs:
A =3D (8, 60o)P ,   B =3D (4, 0o)P
C =3D (4, -45o)P ,=   D =3D (-4, 135)P
Negative r means we go in the opposite direction of q.
135o
Hence,  D =3D (-4, 135o)P is the = same as C =3D (4, -45o)P. 
D =3D (-4, 135o)P
The conversion rules b= etween the polar coordinates and the rectangular coordinates from are the s= ame = as before= .
------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058_image043.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPgALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwA5 AAUAgAAAAAAAAAIbhI+py70RDpy02ouzhmD7D17OSJbKZKZqEx0FADs= ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058_image044.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgCoAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAG AKMAgAAAAAAAAAJSBBJmqMjnoEtQ1UZXvtq+jnnbx5XkOabiGrbga6puzMJoTc+3Ltt97sPtgjzh bwhMIpfHpvFZjBKnSqe0CqUys9jr1mvVhrti7rcMNo/P5Pa3AAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058_image045.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOQBjAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAA2 AGAAgQAAAAAAALvg4/8AAALOhI+pyxv/mpxUBYFxqLyjm2WbR05gKIzlmpyoysYAOISwvF5DLeLy vuv5WMAdZIgDIpeDpfMJjUqnu+mqavVgs50tl+L9TsLisvmMTpOaaga5/Yaz24i42p6e0/f8vj+p t4eHNmhWaBhId/jH2Oj4eLYoJslFWZko14gJydnp6WSZFUq1eVdKeIr6ucraeqWZisg4KkVbG2vo qrvLW4c7+Qv8ZxtF/GTcm6xcORv8hczkfJl5UFSUYB2YrZ2NrWSw7X3tay0+Xl0u+M3nVQAAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058_image046.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUwADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAABQ AAIAgAAAAAAAAAILjI+py+0Po5y0rgIAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058_image047.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOAAtAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAA2 ACgAgQAAAAAAALvg4/8AAAJzhI+pyw0Bo5u0PoFzsLyHDG7dOH2gNpCqYp5CMKTr3J5bLM9kjYkA rtvZErHgLuIzKpfMpnNSfOqi0hWwqrpiR9St9wsOf7tiB7nM0KLT5zUi547LpfD5oT3Hy9X2n95d 1ydIEbj3h4ZTuJaouMjXd3hQAAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0058_image048.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhUQBRAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAQABQBM AEwAgAAAAP8AAAKdTICpy+0P4zNG2oszolX7n3HgSEJiiaIH0KXuC8fyPLf03dj4vq38DwwKf7ph rGh8IZPMppO5fIKiUg+1is1qNddtpOud+MLkshlsVqDTPbb77Vyz5Wk6/I6fjvMLe3nPFyg44kdW GAY4qLj4lRh46OXIyCiZB7l1qVU5Obh5l5kFiuXJyUf6JlqVKnVa+sm5+hQb5+raOndbl3tWAAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0059.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
Conversion Rules:
Let (x, y= ) and (r, q)P = be the rectangular and polar coordinates of the same point P,
------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0060.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
Conversion Rules:
Let (x, y= ) and (r, q)P = be the rectangular and polar coordinates of the same point P, then:
A =3D(x, y)=3D (r, q)P<= /div>
r
q
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Polar Coordinate= s
Conversion Rules:
Let (x, y= ) 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) <= /div>
A =3D(x, y)=3D (r, q)P<= /div>
r
q
x =3D r*cos(q)
y =3D r*sin(q)
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Polar Coordinate= s
Conversion Rules:
Let (x, y= ) 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) <= /div>
A =3D(x, y)=3D (r, q)P<= /div>
r
q
x =3D r*cos(q)
y =3D r*sin(q)
Converting from polar to rectangular coordinates
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Polar Coordinate= s
Conversion Rules:
Let (x, y= ) 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) <= /div>
A =3D(x, y)=3D (r, q)P<= /div>
r
q
x =3D r*cos(q)
y =3D r*sin(q)
r =3D x2 = + y2, or r =3D Öx + y2 =
Converting from polar to rectangular coordinates
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Polar Coordinate= s
Conversion Rules:
Let (x, y= ) 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) <= /div>
A =3D(x, y)=3D (r, q)P<= /div>
r
q
x =3D r*cos(q)
y =3D r*sin(q)
r<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D x2 + y2, or r =3D &Ou= ml;x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 + y2 
tan(q) =3D y/x
Converting from polar to rectangular coordinates
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Polar Coordinate= s
Conversion Rules:
Let (x, y= ) 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) <= /div>
A =3D(x, y)=3D (r, q)P<= /div>
r
q
x =3D r*cos(q)
y =3D r*sin(q)
Converting from rectangu= lar to polar coordinates.
Solving for q d= epends on the position of the point.
r<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 =3D x2 + y2, or r =3D &Ou= ml;x<= span lang=3DEN-US style=3D'font-size:103%;position:relative;top:-.45em;mso-text= -raise: 30%;mso-fareast-language:ZH-CN'>2 + y2 
tan(q) =3D y/x
Converting from polar to rectangular coordinates
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Polar Coordinate= s
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, 135)P 
------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0071.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
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, 135)P 
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Polar Coordinate= s
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, 135)P 
For A =3D (4, 60)P ,  
(x, y) =3D (4*cos(60), 4*sin(60))
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file:///C:/D9125D13/7PolarCoordinates_files/slide0079_image097.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdhSwBQAHcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAASwBQAID///8B AgMCVYSPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2PJwoAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0072.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
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, 135)P 
For A =3D (4, 60)P ,  
(x, y) =3D (4*cos(60), 4*sin(60))
         =3D  (2, 2Ö3)
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file:///C:/D9125D13/7PolarCoordinates_files/slide0072_image099.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdhSwBQAHcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAASwBQAID///8B AgMCVYSPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2PJwoAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0073.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
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, 135)P 
For A =3D (4, 60)P ,  
(x, y) =3D (4*cos(60), 4*sin(60))
         =3D  (2, 2Ö3)
For B =3D (8, 0)P , (x, y) =3D (8, 0)
(8, 0)
(2, 2Ö3)=
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file:///C:/D9125D13/7PolarCoordinates_files/slide0073_image101.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdhSwBQAHcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAASwBQAID///8B AgMCVYSPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2PJwoAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0074.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
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, 135)P 
For A =3D (4, 60)P ,  
(x, y) =3D (4*cos(60), 4*sin(60))
         =3D  (2, 2Ö3)
For B =3D (8, 0)P , (x, y) =3D (8, 0)
For C and D, 
(x, y) =3D (4cos(-45), 4sin(-45))
=3D (-4cos(135), -4sin(135))
(8, 0)
(2, 2Ö3)=
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file:///C:/D9125D13/7PolarCoordinates_files/slide0074_image103.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdhSwBQAHcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAASwBQAID///8B AgMCVYSPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2PJwoAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0075.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Polar Coordinate= s
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, 135)P 
For A =3D (4, 60)P ,  
(x, y) =3D (4*cos(60), 4*sin(60))
         =3D  (2, 2Ö3)
For B =3D (8, 0)P , (x, y) =3D (8, 0)
For C and D, 
(x, y) =3D (4cos(-45), 4sin(-45))
=3D (-4cos(135), -4sin(135)) =3D (2Ö2, -2Ö2)
(8, 0)
(2, 2Ö3)=
(2Ö2, -2<= /b>Ö2)=
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file:///C:/D9125D13/7PolarCoordinates_files/slide0075_image105.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdhSwBQAHcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAASwBQAID///8B AgMCVYSPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2PJwoAOw== ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0040.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Example: Convert the following
to polar coordinate:
A =3D (-4, 6),   B =3D (-8, 0)= , 
C =3D (-3, -2)
Polar Coordinate= s
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhMwBIAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAUABAAt AEMAgAAAAAAAAAKLDI6py+1hnJwURoozsFD7Z33iwo3mEXXnqa5m64pwrM10Zt/VpWN5DwLWeMJg cXdEJo3Lxq+5IUJR0mnUSqpCn03u0nsEh7Vb8td8xirEQnYbnXT35HP4WI2g6/Q3fsz/Z1cEuEJY KPiGR6W4yGj4ggj0COkYWcd4pTg5silj2fcJitnpQVoaShNQAAA7 ------=_NextPart_01C85853.4139DC60 Content-Location: file:///C:/D9125D13/7PolarCoordinates_files/slide0081.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Polar Coordinate
Example: Convert the following
to polar coordinate:
A =3D (-4, 6),   B =3D (-8, 0)= , 
C =3D (-3, -2)
For A, r =3D Ö16+36 =3D Ö52
Polar Coordinate= s
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Example: Convert the following
to polar coordinate:
A =3D (-4, 6),   B =3D (-8, 0)= , 
C =3D (-3, -2)
For A, r =3D Ö16+36 =3D Ö52
q =3D arccos(-4/Ö52) » 123o &#= 13;
Polar Coordinate= s
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