BC 3
A Re-View of Polar Coordinates
Name:
We’ve seen polar coordinates previously, and it’s now time to take another look. We begin with a quick
review of the basic set-up. Later, we’ll extend the earlier work to look at the calculus of polar graphs.
We begin with point O, the pole, and a polar axis.
O
pole
polar axis
We label a point P(r, ) where r is the distance from P to the pole and  is the angle formed between the
positive x-axis and the ray that extends to the point. To plot points, it is often easiest to find the ray
determined by the angle and then go out a distance r from the pole, even though the coordinates are not
given in this order.
Examples:
P(2, 30)
Q(–2, 30)
 3 
R 1, 
 4 
3  
S , 
2 2
P
R
O
Q
1.
Explain how to plot a point if r is negative.
2.
Plot and label the following points:
 
 3 
a. A  2, 
b. B 1, 
 4 
 6




c.
C  3,  
d. D  3, 
2
4




 5 
e. E  2 ,  
f.
F 2, 
3
3 


2
S
1
3.
What is significant about points E and F in #2? Can this situation occur with
rectangular (x-y) coordinates?
4.
Find three alternative coordinate pairs for the following point:
 7 
or (
,
) or (
,
) or
 3,

 6 
IMSA
1
Polar 1.1
(
,
2
3
)
F14
5.
We need to examine the relationship between the ordered pairs (x, y) and (r, ), which are
usually distinguished by context. Based on the information in the figure below at the left:

Rectangular
(x, y)
(x, y)
Polar:
(r, θ)
a.
cos  =

x=
b.
sin  =

y=
c.
r2 =
(in terms of x and y)
d.

tan  =
(in terms of x and y)
r
y
θ
x
6.
Change the following from polar to rectangular coordinates, finding exact values if possible.

 2 

a.
b.
c.
(4, 212)
 6,

 2, 
4
 3 

7.
Change the following from rectangular to polar coordinates. (A sketch may be helpful in finding
the angle.)
a.
(3, –3)
b.
(–2, 2 3 )
c.
(–3, 5)
8.
Change the following equation from polar to rectangular form.
a.
9.
b.
r  4csc
Change the following equation from rectangular to polar form.
a.
IMSA
r  4cos
x2  y 2  2 y  0
b.
Polar 1.2
y  2x 1
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10.
Write an equation describing the following graphs:
a) _____________________
11.
b) ___________________
a. Sketch the graph of the limacon r  2  4sin( ), 0    2 .
(Label intercepts using polar coordinates)
b. Find all values of  , 0    2 such that r = 0.
IMSA
Polar 1.3
F14