9'6 Polar Coordinates

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9.6 Polar Coordinates
To change from polar coordinates to rectangular
coordinates, use the formulas x rcos? and y
rsin? To change from rectangular coordinates
to polar coordinates, use the formulas and
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9.6 Polar Coordinates
Limacons r a b cos


r 2 3cos? r 2 2 cos? r 3 2cos?
r 2 cos? (a lt b) (a b) (b
lt a lt 2b) (a ?2b) Limacon with
Cardioid Dimpled Convex
a Loop
Limacon Limacon
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9.6 Polar Coordinates
Note Changing from cos? to sin ? will cause the
graph to be vertical. For example the graph of r
2 3 sin ? is below. Also negating the
trig function will cause the graph to go in the
opposite direction. For example the graph of r
2 3 sin ? is below.
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9.6 Polar Coordinates
Rose Curves   r a cos b or r a sin b  
length of the petal   Examples r
3 sin 5? r 2 sin 4? r 3 cos 5?    
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9.6 Polar Coordinates
Circles  Examples of circle equations r a,
r a cos?, r a sin?, r a sin (?
-?) (Note that a sin (? -?) a sin ?cos? - a
cos?sin? Asin? Bcos?, the 4th example in the
list below.)    r 3 r 3 cos ? r
-4 sin ? r 2cos? 3 sin?    


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9.6 Polar Coordinates
Lemniscates   Lemniscates look like figure 8s
and they have the following equations r2
a2sin2? or r2 a2 cos2?.   r2 4
sin2? r2 4 cos2?