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(b1) (5,−5π/6, π/3) is already in the canonical set.
(b2) x = r cos p sin h = 5 cos(π/3) sin(−5π/6) = 5(1/2)(−1/2) = −5/4
y = −r sin p = −5 sin(π/3) = −5(
√
3/2) = −(5
√
3)/2
z = r cos p cos h = 5 cos(π/3) cos(−5π/6) = 5(1/2)(−
√
√
3/2) = −(5
3)/4
√
√
3)/4)
so (x, y, z) = (−5/4,−(5
3)/2,−(5
(c1) (2,−π/6, π) =⇒ (2, 5π/6, 0)
(c2) x = r cos p sin h = 2 cos(0) sin(5π/6) = (2)(1)(1/2) = 1
y = −r sin p = −2 sin(0) = (−2)(0) = 0
z = r cos p cos h = 2 cos(0) cos(5π/6) = (2)(1)(−
√
3/2) = −
√
3
so (x, y, z) = (1, 0,−
√
3)
(d1) (8, 9π/4, π/6) =⇒ (8, π/4, π/6)
(d2) x = r cos p sin h = 8 cos(π/6) sin(π/4) = 8(
√
√
2/2) = 2
√
6
3/2)(
y = −r sin p = −8 sin(π/6) = −8(1/2) = −4
z = r cos p cos h = 8 cos(π/6) cos(π/4) = 8(
√
3/2)(
√
√
2/2) = 2
6
√
√
so (x, y, z) = (2
6,−4, 2
6)
√
√
3)
2
+ (−
√
√
√
16 = 4
10.
(a) r =
x
2
+ y
2
+ z
2
=
(
2)
2
+ (2
2)
2
=
2 + 12 + 2 =
h = arctan(x/z) = arctan(−
√
√
2) = arctan(−1) = 135
o
, given the location of
2/
(x, z)
p = arcsin(−y/r) = arcsin(−(2
√
3)/4) = arcsin(−
√
3/2) = −60
o
so (r, h, p) = (4, 135
o
,−60
o
)
√
3)
2
+ 6
2
+ (−4)
2
=
√
√
(b) r =
x
2
+ y
2
+ z
2
=
(2
12 + 36 + 16 =
64 = 8
√
3)/4) = arctan(−
√
3/2) = 139.11
o
, given the loca-
h = arctan(x/z) = arctan(−(2
tion of (x, z)
p = arcsin(−y/r) = arcsin(−6/8) = arcsin(−3/4) = −48.59
o
so (r, h, p) = (8, 139.11
o
,−48.59
o
)
(c) r =
√
1 + 1 + 1 =
√
3
h = arctan(x/z) = arctan((−1)/(−1)) = arctan(1) = −135
o
, given the location of
(x, z)
p = arcsin(−y/r) = arcsin(1/
x
2
+ y
2
+ z
2
=
(−1)
2
+ (−1)
2
+ (−1)
2
=
√
3) = 35.26
o
√
3,−135
o
, 35.26
o
)
so (r, h, p) = (
√
2
h = arctan(x/z) = arctan(2/4) = arctan(1/2) = 26.57
o
, given the location of (x, z)
p = arcsin(−y/r) = arcsin((2
√
√
√
32 = 4
(d) r =
x
2
+ y
2
+ z
2
=
2
2
+ (−2
3)
2
+ 4
2
=
4 + 12 + 16 =
√
3)/(4
√
√
√
2)) = 37.76
o
2)) = arcsin(
3/(2
√
2, 26.57
o
, 37.76
o
)
so (r, h, p) = (4
(−
√
3)
2
+ (−
√
√
2)
2
=
√
√
(e) r =
x
2
+ y
2
+ z
2
=
3)
2
+ (2
3 + 3 + 8 =
14
h = arctan(x/z) = arctan(−
√
√
2)) = −31.48
o
, given the location of (x, z)
3/(2
√
3/
√
14) = 27.58
o
p = arcsin(−y/r) = arcsin(
√
14,−31.48
o
, 27.58
o
)
so (r, h, p) = (
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