Problem Set (Polar Coordinates and Polar Equations)
1,
Convert the following rectangular coordinates to polar coordinates.
2.
Convert the following polar coordinates to rectangular coordinates.
3.
Convert the following Cartesian equations to polar equations.
a) a)ß "!b
a) a(ß 1b
b) a • "ß )b
b) a'ß $1Î%b
a) B# € C# œ "&
c) a • &ß • *b
c) a • $ß $1Î#b
b) B# € C# œ (C
c) C œ '
4.
Convert the following polar equations to Cartesian equations.
a) < œ "!
5.
b) < œ "!=38 )
c) < œ &=/- )
Sketch the graphs of the polar equations.
a) < œ ) where )
d) < œ =38 #)
!
b) < œ &
c) < œ ' -9= )
e) < œ # • #-9= )
.C
at the point on the graph
.B
where ) œ 1Î', ) œ 1Î%, and ) œ 1Î#Þ
6.
If < œ ' =38 ) find
7.
Use your calculator to find these areas.
a) Find the area of one leaf of the graph of < œ '=38 $)Þ
b) Find the area of the cardioid given by < œ $ • $-9= )Þ
Solutions
1.
a) < œ È'% € "!! œ È"'% œ 12.8062
"!
&
) œ >+8•"
œ >+8•" œ 0.896055
)
%
Ans: a"#Þ)"ß !Þ)*'" <b
b) < œ È" € '% œ È'& œ 8.06226
) is in quadrant 2 so
)
) œ 1 • E<->+8Œ • œ 1.69515
"
Ans: a)Þ!'#ß "Þ'*& <b
c) < œ È#& € )" œ È"!' œ 10.2956
) is in quadrant 3 so
*
) œ 1 € E<->+8Œ • œ 4.20529
&
Ans: a"!Þ$!!ß %Þ#!& <b
2.
a) Ans: a • (ß !b
È#
$1
œ'†
œ • $È #
%
#
È#
$1
C œ '=38
œ •'†
œ $È #
%
#
Ans: Š • $È#ß $È#‹
b) B œ '-9=
c) Ans: a!ß $b
3.
a) B# € C# œ "& Ê <# œ "& or better < œ È"& Þ
b) B# € C# œ (B
Ê
< œ (-9= )Þ
<# œ (<-9= ) or better
c) C œ ' Ê < =38 ) œ ' Ê < œ
4.
a) < œ "!
Ê
'
Ê < œ ' -=- )Þ
=38 )
ÈB# € C# œ "! or better B# € C# œ "!! Þ
b) B# € C# œ (C Ê <# œ (<=38 ) Ê < œ (=38 )Þ
<
&
c) < œ &=/- ) Ê < œ & †
Ê "œ
Ê B œ &Þ
B
B
5.
a) < œ ) where )
!
b) < œ & has a graph which is a circle of radius 5 centered
at the origin.
c) < œ '-9= ) has a graph which is a circle of diameter 6
centered at a$ß !bÞ
d) < œ =38 #) has a graph with 4 leaves.
e) < œ # • #-9= ) has a graph which is a cardioid.
6.
< œ '=38 )
B œ < -9= ) œ '=38 ) -9= )
.B
œ '-9= ) -9= ) € '=38 )a • =38 )b œ
.)
'a-9=# ) • =38# )b œ '-9= #)Þ
C œ < =38 ) œ '=38 ) =38 ) œ '=38# )
.C
œ "#=38 ) -9= )
.)
.C
.CÎ. )
"#=38 ) -9= )
# =38 ) -9= )
=38 #)
œ
œ
œ
œ
œ
.B
.BÎ. )
'-9= #)
-9= #)
-9= #)
>+8 #)Þ
1
1
.C
œ >+8Š# † ‹ œ >+8 œ È$ Þ
º
.B )œ1Î'
'
$
1
1
.C
œ >+8Š# † ‹ œ >+8 œ _Þ
º
.B )œ1Î%
%
#
Getting _ means the tangent line is vertical.
1
.C
œ >+8Š# † ‹ œ >+8 1 œ !Þ
º
.B )œ1Î#
#
Getting 0 means the tangent line is horizontal.
7.
a) The graph of < œ '=38 $) has 3 leaves as shown below.
We can find the area of 1/2 of one leaf of < œ ' =38 $).
Note that 1/2 of one leaf results when ! Ÿ ) Ÿ 1Î'Þ
" 1Î'
Therefore the area of one leaf is # † (
a'=38 $)b# . ) œ
# !
(
!
1Î'
$'=38# $). ) œ 9.42478 Þ
b) The graph of C œ $ • $-9= ) is shown below.
1/2 of this cardioid results when ! Ÿ ) Ÿ 1Þ
" 1
Therefore the area is # † ( a$ • $-9= )b# . ) œ 42.4115 Þ
# !