patino (mp25752) – Assignment2 – luecke – (57510)
1
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printout
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have
15
questions.
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before answering.
001
10.0 points
Locate the points given in polar coordinates
by
P
parenleftBig
2
,
1
6
π
parenrightBig
,
Q
parenleftBig

1
,
1
2
π
parenrightBig
R
parenleftBig
2
,
2
3
π
parenrightBig
,
among
2
4

2

4
2
4

2

4
1.
P
:
Q
:
R
:
2.
P
:
Q
:
R
:
correct
3.
P
:
Q
:
R
:
4.
P
:
Q
:
R
:
5.
P
:
Q
:
R
:
6.
P
:
Q
:
R
:
Explanation:
To convert from polar coordinates to Carte
sian coordinates we use
x
=
r
cos
θ ,
y
=
r
sin
θ .
For then the points
P
parenleftBig
2
,
1
6
π
parenrightBig
,
Q
parenleftBig

1
,
1
2
π
parenrightBig
R
parenleftBig
2
,
2
3
π
parenrightBig
,
correspond to
P
:
Q
:
R
:
in Cartesian coordinates.
keywords: polar coordinates, Cartesian coor
dinates, change of coordinates,
002
10.0 points
Which, if any, of
A.
(4
, π/
3)
,
B.
(

4
,
7
π/
6)
,
C.
(4
,
13
π/
6)
,
are polar coordinates for the point given in
Cartesian coordinates by
P
(2
,
2
√
3)?
1.
all of them
2.
A only
correct
3.
C only
4.
A and B only
5.
A and C only
6.
B only
7.
none of them
8.
B and C only
Explanation:
To convert from Cartesian coordinates to
polar coordinates we use the relations:
x
=
r
cos
θ ,
y
=
r
sin
θ ,
so that
r
2
=
x
2
+
y
2
,
tan
θ
=
y
x
.
For the point
P
(2
,
2
√
3) in Cartesian co
ordinates, therefore, one choice of
r
and
θ
is
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patino (mp25752) – Assignment2 – luecke – (57510)
2
r
= 4 and
θ
=
π/
3, but there are equivalent
solutions for
r <
0 as well as values of
θ
dif
fering by integer multiples of
π
. For the given
choices we thus see that
A.
TRUE: solution noted already.
B.
FALSE:
θ
incorrect.
C.
FALSE: differs from
π/
6 by 2
π
.
003
10.0 points
Which, if any, of
A.
(2
,
5
π/
6)
,
B.
(2
,
17
π/
6)
,
C.
(

2
,
5
π/
6)
,
are polar coordinates for the point given in
Cartesian coordinates by
P
(

√
3
,
1)?
1.
none of them
2.
C only
3.
B and C only
4.
A only
5.
all of them
6.
A and C only
7.
B only
8.
A and B only
correct
Explanation:
To convert from Cartesian coordinates to
polar coordinates we use the relations:
x
=
r
cos
θ ,
y
=
r
sin
θ ,
so that
r
2
=
x
2
+
y
2
,
tan
θ
=
y
x
.
For the point
P
(

√
3
,
1) in Cartesian co
ordinates, therefore, one choice of
r
and
θ
is
r
= 2 and
θ
= 5
π/
6, but there are equivalent
solutions for
r <
0 as well as values of
θ
dif
fering by integer multiples of
π
. For the given
choices we thus see that
A.
TRUE: solution noted already.
B.
TRUE: differs from 5
π/
6 by 2
π
.
C.
FALSE:
θ
incorrect.
004
10.0 points
A point
P
is given in Cartesian coordinates
by
P
(1
,
1).
Find polar coordinates (
r, θ
) of
this point with
r <
0 and 0
≤
θ <
2
π
.
1.
parenleftBig

√
3
,
5
π
4
parenrightBig
2.
parenleftBig

√
2
,
3
π
4
parenrightBig
3.
parenleftBig

√
3
,
7
π
4
parenrightBig
4.
parenleftBig

√
2
,
5
π
4
parenrightBig
correct
5.
parenleftBig

√
2
,
7
π
4
parenrightBig
6.
parenleftBig

√
3
,
3
π
4
parenrightBig
Explanation:
Since the relationship between Cartesian
coordinates and polar coordinates is
x
=
r
cos
θ ,
y
=
r
sin
θ ,
the point
P
(1
,
1) in Cartesian coordinates can
be given in polar coordinates as
P
parenleftBig

√
2
,
5
π
4
parenrightBig
,
005
10.0 points
Find a Cartesian equation for the curve
given by the polar equation
r
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 Spring '09
 Lueke
 Cartesian Coordinate System, Sin, Cos, Polar coordinate system

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