M408D SPRING 2015 HOMEWORK #7 - cheatham(sc36975 HW07...

cheatham (sc36975) – HW07 – um – (53890) 1
This print-out should have 20 questions.
Multiple-choice questions may continue on
the next column or page – find all choices
before answering.
001 10.0 points
Locate the points given in polar coordinates
by
P 4 ,
2
3
π , Q − 3 ,
1
2
π R 4 ,
1
3
π ,
among
2 4 − 2 − 4
2
4
− 2
− 4
1.
P :
Q :
R :
correct
2.
P :
Q :
R :
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 4 ,
2
3
π , Q − 3 ,
1
2
π R 4 ,
1
3
π ,
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 / 3) ,
C. ( − 4 , 4 π / 3) ,
are polar coordinates for the point given in
Cartesian coordinates by P (2 , 2
√
3)?
1. all of them correct
2. B and C only
3. B only
4. C only
5. none of them
6. A and B only
7. A only
8. A 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

cheatham (sc36975) – HW07 – um – (53890) 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. TRUE: di ff ers from π / 3 by 2 π .
C. TRUE:
− 4 cos(4 π / 3) = 2 , − 4 sin(4 π / 3) = 2
√
3 .
003 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 .
Which one of the following shaded-regions
in the plane consists of all points whose polar
coordinates satisfy the inequalities
3 < r ≤ 4 ,
1
3
π ≤ θ ≤
5
6
π ?
2 4
6.
2
correct
4

Explanation: