Honors Precalculus
Name___________________________________
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Polar Comp Book Problems
Plot the point with the given polar coordinates.
(
1) (-3, 90°)
2) 2, 90°
120°
3p
4
)
60°
150°
2p
3
30°
p
2
p
3
5p
6
180°
1
2
3
4
p
6
0°
p
210°
1
2
3
4
0
330°
240°
7p
6
300°
11p
6
270°
4p
3
5p
3
3p
2
( )
3) (3, -165°)
4) -4,
90°
120°
p
6
60°
150°
2p
3
30°
p
2
p
3
5p
6
180°
1
2
3
4
p
6
0°
p
210°
1
2
3
4
0
330°
240°
7p
6
300°
11p
6
270°
4p
3
3p
2
5p
3
Worksheet by Kuta Software LLC
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Find all pairs of polar coordinates that describe the same point as the provided polar
coordinates.
( )
5) 2,
6) (4, 195°)
5p
6
Convert each pair of polar coordinates to rectangular coordinates.
( )
7) 3,
8) (4, 150°)
3p
2
Convert each pair of rectangular coordinates to polar coordinates where r > 0 and 0 £ q < 2p .
10) (0, 1)
9) (-2, 2 3 )
Two points are specified using polar coordinates. Graph the points. Find the distance between
the points.
( )( )
11) 2,
( )(
3p
p
, 3,
2
4
2p
3
12) 2,
p
2
2p
3
p
3
5p
6
1
2
7p
6
3
4
0
3p
2
p
3
p
6
p
1
2
7p
6
11p
6
4p
3
p
2
5p
6
p
6
p
5p
, 1, 0)
6
5p
3
3
4
0
11p
6
4p
3
3p
2
5p
3
Worksheet by Kuta Software LLC
-2-
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Convert each equation from rectangular to polar form.
13) x =
y2
5
14) y = -x
2
2
15) ( x - 1) + ( y + 1) = 2
16) y = 4x
Convert each equation from polar to rectangular form.
17) r = 5tan q sec q
18) r = 4sin q
2
19) r = 5sec (2q )
20) tan q = 4
Consider each polar equation over the given interval. Classify the curve; determine if the graph
is symmetric with respect to the origin, polar axis, and line q =p /2; find the values of q where r
is zero; find the maximum r value and the values of q where this occurs; and sketch the
graph.
21) r = 4cos (2q ), 0 ≤ q < 2p
2p
3
p
2
p
3
5p
6
p
6
p
1 2 3 4 5 6 7
7p
6
0
11p
6
4p
3
3p
2
5p
3
Worksheet by Kuta Software LLC
-3-
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2
22) r = 36cos (2q ), 0° ≤ q < 360°
90°
120°
60°
150°
30°
180°
0°
1 2 3 4 5 6 7
210°
330°
240°
300°
270°
23) r = 2 + 4cos q , 0 ≤ q < 2p
2p
3
p
2
p
3
5p
6
p
6
p
1 2 3 4 5 6 7
7p
6
0
11p
6
4p
3
3p
2
5p
3
24) r = 3 - 3sin q , 0° ≤ q < 360°
90°
120°
60°
150°
30°
180°
1 2 3 4 5 6 7
210°
0°
330°
240°
300°
270°
Worksheet by Kuta Software LLC
-4-
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Answers to Polar Comp Book Problems
1)
2)
120°
3)
90°
150°
30°
180°
1 2 3 4
210°
p
6
1 2 3 4
7p
6
11p
6
4p
3
270°
(
) (
)
5p
6
240°
300°
5p
6
p
( )
9) 4,
4p
3
23)
3
2p
3
5p
6
{
{
p
7p
6
3p
2
3p
2
5p
3
p
2
13) r = 5cot q csc q
10) 1,
180°
}
210°
}
20) y = 4x
Lemniscate
Symmetric about the origin,
30° polar axis, and line q = 180°
r = 0 when q = {45°, 135°, 225°, 315°}
0° r = 6 when q = {0, 180°}
2 4 6
150°
330°
240°
300°
270°
24)
120°
90°
}
Cardioid (Limaçon)
Symmetric about the line q = 180°
30° r = 0 when q = {90°}
r = 6 when q = {270°}
0°
2 4 6
60°
150°
180°
210°
11p
6
4p
3
11p
6
( )
2p
3
2
{
7p
6
7p
6
0
6) (4, 195° + 360n°) and (-4, 15° + 360n°)
where n is an integer
13 + 6 2 » 4.635
12) 5 + 2 3 » 2.909
3p
15) r = 2cos q - 2sin q
16) tan q = 4
14) q =
4
2
2
x2
19) x 2 - y 2 = 5
18) x + ( y - 2) = 4
17) y =
5
90°
p
22)
21)
120°
2p
60°
p
Rose
p Symmetric about the origin,
6 polar axis, and line q = p
2
0
p 3p 5p 7p
2 4 6
r = 0 when q = ,
,
,
4 4 4 4
11p
p
3p
r = 4 when q = 0, , p,
6
2
2
5p
3p
3
2
p
p
2
Looped limaçon
3
p Symmetric about the polar axis
6 r = 0 when q = 2p , 4p
3 3
0 r = 6 when q = {0}
2 4 6
1 2 3 4
4p
3
270°
11)
3
p
6
p
330°
5p
3
3p
2
5p
5p
+ 2np and -2,
+ (2n + 1)p
6
6
where n is an integer
(
7) 0, -3)
8) (-2 3, 2)
5) 2,
210°
p
3
0°
1 2 3 4
0
p
2
2p
3
60°
30°
180°
p
90°
150°
0°
300°
4)
120°
p
3
5p
6
330°
240°
p
2
2p
3
60°
330°
240°
5p
3
300°
270°
Worksheet by Kuta Software LLC
-5-
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