Gauthier, Joseph – Homework 3 – Due: Sep 15 2007, 3:00 am – Inst: JEGilbert
2
correspond to
P
:
Q
:
R
:
in Cartesian coordinates.
keywords: polar coordinates, Cartesian coor
dinates, change of coordinates,
003
(part 1 of 1) 10 points
Which, if any, of
A.
(

4
,
4
π/
3)
,
B.
(4
, π/
6)
,
C.
(4
,
7
π/
3)
,
are polar coordinates for the point given in
Cartesian coordinates by
P
(2
,
2
√
3)?
1.
B and C only
2.
C only
3.
none of them
4.
A and C only
correct
5.
all of them
6.
A and B only
7.
A only
8.
B 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
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:

4 cos(4
π/
3) = 2
,

4 sin(4
π/
3) = 2
√
3
.
B.
FALSE:
θ
incorrect.
C.
TRUE: differs from
π/
3 by 2
π
.
keywords: T/F, polar coordinaates, Cartesian
coordinates
004
(part 1 of 1) 10 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.
‡

√
2
,
7
π
4
·
correct
2.
‡

√
3
,
5
π
4
·
3.
‡

√
2
,
3
π
4
·
4.
‡

√
2
,
5
π
4
·
5.
‡

√
3
,
3
π
4
·
6.
‡

√
3
,
7
π
4
·
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
‡

√
2
,
7
π
4
·
,
keywords: polar coordinates, Cartesian coor
dinates,
005
(part 1 of 1) 10 points