Precalculus A Worksheet 10-04
Basic Polar Review
Section 9.2
Name _____________________
Date _________ Period ______
Please fill in the blank for the following questions.
1) In polar coordinates, the origin is called the ______ and the positive x-axis is called the
______ _____.
2)
4

 
Another representation in polar coordinates for the point  2,  is  ___,
3
 3

3)


The polar coordinates  2,  are represented in rectangular coordinates by
6


.

 _____, _____  .
Please determine if the following questions are True or False.
4)
The polar coordinates of a point are unique.
True / False
5)
The rectangular coordinates of a point are unique.
True / False
6)
In r ,   , the number r can be negative.
True / False
Match each of the following polar coordinates with either A, B, C, or D as plotted
on the


2
2
graph below.
3
3
90

3
11 

 2, 

6 

8)


10)  2,  
6

11)
7)
5 

 2,

6 

9)
7 

 2,

6 

 11 
12)  2,

6 

 2,  


6

120
4
135
5
6 150
60
A
B
 180
1
2
3
D
C
7 210
6
225
5
240
4
4
3
45
4

30 6
4
0
330 11
6
315
7
300
4
5
3
270
3
2
Plot each point given in polar coordinates, and find other polar coordinates r ,   , of the point
for which . . .
a) r  0 and  2    0
13) E 2,  
b) r  0 and 0    2


14) F  3,  
4

2 

15) G  4, 

3 

c) r  0 and 2    4
2
3
3
120
4
135

2
90

3
60
5
6 150
 180
7 210
6
225
5
240
4
4
3
1
2
45
3

4

30 6
4
0
330 11
270
3
2
315
7
300
4
5
3
6
Given the polar coordinates of each of the following points, find its rectangular coordinates.
 3 
16)  4,
17)  3.1,182 
18)  8.1, 5.2R 

 2 
Given the rectangular coordinates of each of the following points, find its polar coordinates
such that r  0 and 360    0 .
19)
 0,2
20)
 2, 2 3 
21)
 2.3, 0.2
The letters x and y represent rectangular coordinates. Rewrite each of the following
equations using polar coordinates r ,   .
22) x 2  4y
23) 2xy  1
The letters r and  represent polar coordinates. Rewrite each of the following equations
using rectangular coordinates  x , y  .
24) r  sin   1
25) r 
4
1  cos 
Without using a calculator, restate each of the following polar equations using rectangular
coordinates and then graph the equation.
19
26)   120
27) r  2.5
28)  
29) r  0
4
2
3
3
120
4
135

2
90

3
60
5
6 150
 180
1
7 210
6
225
5
240
4
4
3
2
45
3
2
3
3
120
4
135

4
0
330 11
6
315
7
300
4
5
3
270
3
2
 180

2
90

3
60
5
6 150
 180
1
7 210
6
225
5

4
240
4
3
2
45
3
2
3
3
120
4
135
0
330 11
270
3
2
3

4
4
0
330 11
270
3
2

2
90

3
60
6
 180
1
7 210
6
225
5

4
240
4
3
2
45
3
6

3
60
 180
1
7 210
6
225
5
240
4
4
3
45
3
2
2
3
3
120
4
135

4
4
 180
0
330 11
270
3
2
2
3
3
120
4
135

4
315
7
300
4
5
3

3
60
5
6 150

30 6
4

2
90
0
6
 180
1
7 210
6
225
5

4
240
4
3
2
45
3
0
330 11
270
3
2
6
3
38) r cos   3
39)

30 6
4
0
330 11
270
3
2

2
90

3
60
 180
1
7 210
6
225
5

4
240
4
3
2
45
3
6
r  2  sin 

4

30 6
4
0
330 11
270
3
2
315
7
300
4
5
3
Without using a calculator, match each of the following graphs to one of the polar equations
by writing the equation number in the upper left-hand corner of the appropriate graph.
34) r  4
35) r  3cos 
36)
r  3sin 
37) r sin   3

4
315
7
300
4
5
3
5
6 150

30 6
315
7
300
4
5
3
2
45
33) 3r  12cos 
2
3
3
120
4
135

4
4

3
60
1
7 210
6
225
5
240
4
4
3
6
315
7
300
4
5
3

2
90
5
6 150

30 6
32) r csc   4  2
330 11
270
3
2

2
90
5
6 150

30 6
315
7
300
4
5
3
5
6 150

30 6
315
7
300
4
5
3
2
45
2
3
3
120
4
135
31) r  4 sin 

4
4
1
7 210
6
225
5
240
4
4
3
30) r sec  2
2
3
3
120
4
135

3
60
5
6 150

30 6
4

2
90
6
Precalculus A Worksheet 10-04
Basic Polar Review
Section 9.2
Answer Key
Please fill in the blank for the following questions.
1) In polar coordinates, the origin is called the ______
pole and the positive x-axis is called the
axis
polar _____.
______
2)
3)
4

 
Another representation in polar coordinates for the point  2,  is  ___,
2
3
 3


.



The polar coordinates  2,  are represented in rectangular coordinates by
6

    3
    1
x

r
cos



2cos
y

r
sin



2
sin


 

1

3
_____,
_____
.


6
6
Please determine if the following questions are True or False.
4)
The polar coordinates of a point are unique.
True / False
5)
The rectangular coordinates of a point are unique.
True / False
6)
In r ,   , the number r can be negative.
True / False
Match each of the following polar coordinates with either A, B, C, or D as plotted
on the


2
2
graph below.
3
3
90

3
11 

 2, 
A
6 

8)


10)  2,   B
6

11)
7)
5 

 2,
 D
6 

9)
7 

 2,
 A
6 

 11 
12)  2,
D
6 

 2,  

 C
6

120
4
135
5
6 150
60
A
B
 180
1
2
3
D
C
7 210
6
225
5
240
4
4
3
45
4

30 6
4
0
330 11
6
315
7
300
4
5
3
270
3
2
Plot each point given in polar coordinates, and find other polar coordinates r ,   , of the point
for which . . .
a) r  0 and  2    0
13) E 2,  
a)
b)
c)
2,  
 2, 0 
2,3 
b) r  0 and 0    2


14) F  3,  
4

5 

a )  3, 

4 

7 

b )  3,

4 

 11 
c )  3,

 4 
2 

15) G  4, 

3 

5 

a )  4, 

3 

4 

b )  4,

3 

 7 
c )  4,

3 

c) r  0 and 2    4
2
3
3
120
4
135
5
6 150
 180

2
90

3
60
G
F
E
7 210
6
225
5
240
4
4
3
1
2
45
3

4

30 6
4
0
330 11
270
3
2
315
7
300
4
5
3
6
Given the polar coordinates of each of the following points, find its rectangular coordinates.
 3 
16)  4,
17)  3.1,182 
18)  8.1, 5.2R 

 2 
x  3.1 cos 182   3.10
x  8.1cos 5.2R   3.79
 3 
x  4 cos 
y  3.1 sin 182   0.11
0
y  8.1 sin  5.2R   7.16
 2 
 3.10, 0.11 
 3 
 3.79, 7.16
y  4 sin 


4

 2 
 0, 4 
Given the rectangular coordinates of each of the following points, find its polar coordinates
such that r  0 and 360    0 .
19)
 0,2
20)
r  02  22  2
 2, 2 3 
r 
2
0
 2, 90 
  tan 1    90
2
 2
21)

 2 3

2
4
 2 3 
  tan 
  60
 2 
 4, 300 
1
 2.3, 0.2
r 
2
 2.3
2
  0.2   2.31
 0.2 
  4.97
 2.3 
  2.31, 4.97 
  tan 1 
The letters x and y represent rectangular coordinates. Rewrite each of the following
equations using polar coordinates r ,   .
23) 2xy  1
2  r cos    r sin    1
22) x 2  4y
x 2  4  r sin  
r cos  2  4r sin 
r 2 2cos  sin    1
r 2 cos2   4r sin 
r 2 sin 2   1
r 2 cos2   4r sin   0
The letters r and  represent polar coordinates. Rewrite each of the following equations
using rectangular coordinates  x , y  .
25) r 
24) r  sin   1
r 2  r  sin   1 
r 1  cos    4
r 2  r sin   r
x y y  x y
2
2
4
1  cos 
2
2
r  r cos   4
r  4  r cos 
 x2  y2  4  x
x 2  y 2  16  8x  x 2
y 2  8  x  2
Without using a calculator, restate each of the following polar equations using rectangular
coordinates and then graph the equation.
19
26)   120
27) r  2.5
28)  
29) r  0
4
2
2
 0, 0 
x

y

2.5
y  3x
y  x



3
4
2
3
120
135

3
60
2
90
5
6 150
 180
1
7 210
6
225
5
240
4
4
3
2
45
3
3
4

4
0
330 11
6
315
7
300
4
5
3
270
3
2
 180
2
45
3
4

4
4
0
330 11
6
315
7
300
4
5
3
270
3
2
31) r  4 sin 
2
3
120
135

3
60
2
90
5
6 150

30 6
3
30) r sec  2
r  2cos 
 180
1
7 210
6
225
5
240
4
4
3
45
3
2
3
4

4
 180
0
330 11
270
3
2
1
7 210
6
225
5
240
4
4
3
6
315
7
300
4
5
3

3
60
2
90
5
6 150

30 6
4
2
3
120
135
2
45
3

4

30 6
4
330 11
270
3
2
r  4r sin 
33) 3r  12cos 
r  4 cos 
r  2r cos 
x 2  y 2  4y
r 2  2r sin 
r 2  4r cos 
x 2  y 2  2x
x 2   y 2  4y  4   4
x 2  y 2  2y
x 2  y 2  4x
2
x
2
 2x  1   y 2  1
x 2   y  2  4
2
2
 1  y 2  1
2
3
3
120
4
135

2
90

3
60
5
6 150
 180
1
7 210
6
225
5
2
45
3
2
3
3
120
4
135

4
0
330 11
240
4
3
270
3
2
315
7
300
4
5
3

3
60
5
6 150

30 6
4

2
90
6
 180
1
7 210
6
225
5

4
240
4
3
2
45
3
x
x 2   y  1  1
x
2
2
3
3
120
4
135

4
4
315
7
300
4
5
3

2
90

3
60
5
6 150

30 6
0
330 11
270
3
2
x 2   y 2  2y  1   1
6
 180
1
7 210
6
225
5

4
240
4
3
2
45
3
4
270
3
2

2
90
 180
0
330 11
315
7
300
4
5
3
2
 2  y 2  4

3
60
5
6 150

30 6
1
7 210
6
225
5
6

4
240
4
3
2
45
3
38) r cos   3
35
38
37
39)
34
r  2  sin 
36
39

4

30 6
4
0
330 11
270
3
2
315
7
300
4
5
3
Without using a calculator, match each of the following graphs to one of the polar equations
by writing the equation number in the upper left-hand corner of the appropriate graph.
34) r  4
35) r  3cos 
36)
r  3sin 
37) r sin   3
6
 4x  4   y 2  4
2
2
3
3
120
4
135

4
0
315
7
300
4
5
3
32) r csc   4  2
r  2 sin 
x
4
1
7 210
6
225
5
240
4
4
3
2


3
60
2
90
5
6 150

30 6
4
2
3
120
135
6