# Polar Coordinates and Polar Graphs

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```					10.5: Polar Coordinates

Greg Kelly, Hanford High School, Richland, Washington

One way to give someone directions is to tell them to go three blocks East and five blocks South.
Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.

r

A polar coordinate pair


Initial ray

 r , 
determines the location of a point.


Some curves are easier to describe with polar coordinates:

ra

(Circle centered at the origin)

  o

(Line through the origin)

1 r  2

0  


2


More than one coordinate pair can refer to the same point.

2
210o 30o

2,30o     2, 210o    2, 150o 

150o

All of the polar coordinates of this point are:

2,30o  n  360o  

 2, 150

o

 n  360

o



n  0,  1,  2 ...


Tests for Symmetry: x-axis: If (r, ) is on the graph, so is (r, -).

1

r

r  2cos
0



r
-1 1 2



Tests for Symmetry: y-axis: If (r, ) is on the graph, so is (r, -) or (-r, -).
2

r

1

r

r  2sin 
-1

 
0


1





Tests for Symmetry: origin: If (r, ) is on the graph, so is (-r, ) or (r, +) .

2

tan  r cos 

1

r


-2

 

-1

0

1

2

r

-1

-2



Tests for Symmetry: If a graph has two symmetries, then it has all three:

2

r  2cos  2 
-2 -1

1

0

1

2

-1

-2



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