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10.7Polar Coordinates
Adapted by JMerrill, 2011
Polar Coordinate Systems
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You’ve all seen a polar coordinate system (the movies). Polar
coordinates are used in navigation and look like
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The polar coordinate system is formed by fixing a point, O, which is the pole (or origin).
= directed angle Polar axis
r = directed dista
nce
OPole (Origin)
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
r is the directed distance from O to P.
is the directed angle (counterclockwise) from the polar axis to OP.
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The point lies two units from the pole on the
terminal side of the angle
( , ) 2,3
r
.3
3
2,3
33,4
34
2
32
1 2 3 0
3 units from the pole
Plotting Points
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There are many ways to represent the point 2, .3
2
32
1 2 3 0
2,3
52,3
52,3 3
2,
( , ) , 2r r n
We will only use 1 point instead of multiple representations.
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(r, )(x, y)
Polex
y
(Origin)
y
r
x
The relationship between rectangular and polar coordinates is as follows.
The point (x, y) lies on a circle of radius r, therefore,
r2 = x2 + y2.
Coordinate Conversion
• To convert from polar to rectangular:
• x = r cosθ
• y = r sinθ
• To convert from rectangular to polar:
• tanθ =
• x2 + y2 = r2
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yx
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Coordinate Conversion – the Relationship
cosx r cos xr
siny r sin yr
2 2 2r x y tan yx
(Pythagorean Identity)
Example
• Convert to rectangular coordinates
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1cos co3
24 s 42
x r
3sin sin 4 23 2
4 3y r
, 2, 2 3x y
4,3
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Example:
Convert the point (1,1) into polar coordinates.
, 1,1x y
1tan 11
yx
4
2 2 2 21 1 2r x y
set of polar coordinates is ( , ) 2, .4
One r
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Example:Graph the polar equation r = 2cos .
2
32
1 2 3 0
2
0
–2
–1
0
1
20
r
6
3
2
23
56
76
32
116
2
3
3
3
3 The graph is a circle of radius 2 whose center is at
point (x, y) = (0, 1).
Radian modePolar mode
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Each polar graph below is called a Limaçon.
1 2cosr 1 2sinr
–3
–5 5
3
–5 5
3
–3
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Each polar graph below is called a Lemniscate.
2 22 sin 2r 2 23 cos 2r
–5 5
3
–3
–5 5
3
–3
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Each polar graph below is called a Rose curve.
2cos3r 3sin 4r
The graph will have n petals if n is odd, and 2n petals if n is even.
–5 5
3
–3
–5 5
3
–3
a
a
