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Polar Coordinate System
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Polar
and Cartesian coordinates relations |
Conversion from polar
to rectangular coordinates |
Conversion from
rectangular to polar coordinates |
Polar coordinates of a point |
Equation
of a line in polar form |
Lines
parallel to the axes, horizontal and vertical lines |
Lines
running through the origin or pole (radial lines) |
Polar
equation
of a line |
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Polar coordinate system
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The
polar coordinate system is a two-dimensional coordinate system
in which each point P
on a plane is determined by the length of its position vector r
and the angle q
between it and the positive direction of the x-axis,
where 0 <
r
< + oo
and 0
<
q
< 2p. |
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Polar
and Cartesian coordinates relations, |
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Note,
since the inverse tangent function (arctan or tan-1)
returns values in the range -p/2
< q <
p/2, then |
for points lying in the 2nd or 3rd quadrant |
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and for points lying in the 4th quadrant |
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Example:
Convert Cartesian coordinates
(-1,
-Ö3)
to polar coordinates. |
Solution: |
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and since the point lies in the 3rd quadrant,
then |
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Equation
of a line in polar form |
Lines
parallel to the axes, horizontal and vertical lines |
Lines
parallel to the y-axis |
A
vertical line, x
=
c is
represented by the equation |
r
cosq
=
c
or |
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Lines
parallel to the x-axis |
A
horizontal line, y
=
c is
represented by the equation |
r
sinq
=
c
or |
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Lines
running through the origin or pole (radial lines) |
The
equation of a line through the origin or pole that makes an angle
a with the positive
x-axis |
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is
represented by the equation |
q
=
a |
in polar coordinates. |
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Polar
equation
of a line |
As |
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the
polar equation of a line |
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where
p
is the distance of the line from the pole O
and j
is the angle that the segment p
makes with the polar axis. |
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Example:
Write polar equation of the
line passing through points (-4,
0)
and (0, 4). |
Solution:
Using polar equation of a line |
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Proof,
use of Cartesian to polar
conversion formulas. |
The
intercept form of the line |
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or |
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-x
+ y =
4,
y
=
x
+ 4 |
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-r
cosq
+ r sinq
=
4,
r (sinq
-
cosq
) =
4, |
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Pre-calculus
contents B
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