Adapted by
JMerrill, 2011

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 ).
The polar axis is the ray constructed from O .
Each point P in the plane can be assigned polar coordinates ( r ,

).
P = ( r ,

)
Pole (Origin)
O

= directed angle Polar axis r is the directed distance from O to P .
 is the directed angle (counterclockwise) from the polar axis to OP .
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Plotting Points
 
3 terminal side of the angle
 

3
.

2
 

3
 
3


3,
 3

4

   3

4
3 units from the pole
3

2
1 2 3
0
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There are many ways to represent the point
 
3
.

2

2,
 5

3

 
3


2,
 5

3

   r ,
 
2 n
 

 
3
1 2 3
0
We will only use 1 point instead of multiple representations.
3

2
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5

The relationship between rectangular and polar coordinates is as follows.
y
The point ( x , y ) lies on a circle of radius r, therefore, r 2 = x 2 + y 2 .
( x , y )
( r ,

) r y
Pole

(Origin) x x
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• To convert from polar to rectangular:
• x = r cosθ
• y = r sinθ
• To convert from rectangular to polar:
• tanθ = y x
• x 2 + y 2 = r 2
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Coordinate Conversion – the Relationship tan
  y x cos
  r x sin
  r y x
 r cos
 y
 r sin
 r
2  x
2  y
2
(Pythagorean Identity)
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  y
 r x
 r sin
 cos


4
 sin
 
4 co s
 


3
 
3

4
 


4

2, 2 3

2
3

2
 
2 3
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Example:
Convert the point (1,1) into polar coordinates.
tan
   
  x

4
1
1 r
 x
2  y
2 
1
2
1
2
2
One r
 

2,

4

.
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Example:
Graph the polar equation r = 2cos

.
r
2
1
3

0
–1
3

–2
3
0
2
3

0

3

6


6

2
3
5
2

7
3


2
11
6

2



2
3

2
Radian mode
Polar mode
1 2 3
0
The graph is a circle of radius 2 whose center is at point ( x , y ) = (0, 1).
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–5
Each polar graph below is called a
Limaçon
.
r
 r
3 3

5 –5 5
–3 –3
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–5
Each polar graph below is called a Lemniscate .
r
2  2
2 sin 2

3 r
2  2
3 cos 2

3
5
–5 5
–3
–3
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–5
Each polar graph below is called a Rose curve .
r

2cos3

3 r

3sin 4

3 a
5
–5 a
–3 –3
The graph will have n petals if n is odd, and 2 n petals if n is even.
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