February 16, 2004
S. F. Ellermeyer
Name
Instructions.
This exam contains seven problems, but only six of them will be graded. You
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.P
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iteDON
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to grade. In writing your solution to each problem, include su
ﬃ
cient detail and use correct
notation. (For instance, don’t forget to write “
=
” when you mean to say that two things are
equal.) Your method of solving the problem should be clear to the reader (me). If I have to
struggle to understand what you have written, then you might not get full credit for your
solution even if you get a correct answer.
1. Make sure to show all details of your computations here. You may express angles in
terms of radians or degrees (whichever you choose) and you may approximate angles
by rounding to two decimal places.
(a) Find cylindrical coordinates for the point whose rectangular coordinates are
(
x, y, z
)=
(
−
4
,
8
,
−
2)
.
Solution:
r
=
p
x
2
+
y
2
=
q
(
−
4)
2
+(8)
2
=
√
80 = 4
√
5
.
Also,
tan (
θ
)=
y/x
=
−
2
and
θ
is in the second quadrant of the
xy
plane, so
θ
= arctan (
−
2) +
π
≈
2
.
0344
≈
116
.
56
◦
. Cylindrical coordinates of this point are
thus
(
r,
θ
,z
)=
³
4
√
5
,
arctan (
−
2) +
π
≈
116
.
56
◦
,
−
2
´
.
Let us check our answer:
x
=
r
cos (
θ
)
=4
√
5 cos (arctan (
−
2) +
π
)
=
−
4
√
5 cos (arctan (
−
2))
=
−
4
√
5
μ
1
√
5
¶
=
−
4
and
y
=
r
sin (
θ
)
=4
√
5 sin (arctan (
−
2) +
π
)
=
−
4
√
5 sin (arctan (
−
2))
=
−
4
√
5
μ
−
2
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 Fall '10
 Ellermeyer
 Math, Sin, Cos, Polar coordinate system

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