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√
√
(c) x = r cos(θ) = 6 cos(−π/6) = 6 cos(π/6) = 6(
3
y = r sin(θ) = 6 sin(−π/6) = −6 sin(π/6) = −6(−1/2) = −3
so (x, y, z) = (3
3/2) = 3
√
3,−3,−3)
(d) x = r cos(θ) = 3 cos(3π) = 3 cos(π) = 3(−1) = −3
y = r sin(θ) = 3 sin(3π) = 3 sin(π) = 3(0) = 0
so (x, y, z) = (−3, 0, 1)
√
2
θ = arctan(1/1) = 45
o
so (r, θ, z) = (
√
7.
(a) r =
1
2
+ 1
2
=
√
2, 45
o
, 1)
(b) r =
0
2
+ (−5)
2
= 5
θ = −90
o
, since x = 0 and y < 0
so (r, θ, z) = (5,−90
o
, 2)
(c) r =
(−3)
2
+ 4
2
= 5
θ = arctan(4/(−3)) = 126.87
o
so (r, θ, z) = (5, 126.87
o
,−7)
(d) r =
√
0
2
+ 0
2
= 0
θ = 0, since x = 0 and y = 0
so (r, θ, z) = (0, 0,−3)
√
√
8.
(a) x = r sin(φ) cos(θ) = 4 sin(3π/4) cos(π/3) = 4(
2/2)(1/2) =
2
√
2/2)(
√
√
6
y = r sin(φ) sin(θ) = 4 sin(3π/4) sin(π/3) = 4(
3/2) =
z = r cos(φ) = 4 cos(3π/4) = 4(−
√
√
2/2) = −2
2
√
2,
√
√
2)
(b) x = r sin(φ) cos(θ) = 5 sin(π/3) cos(−5π/6) = 5(
so (x, y, z) = (
6,−2
√
3/2)(−
√
3/2) = −15/4
√
3/2)(−1/2) = −5
√
y = r sin(φ) sin(θ) = 5 sin(π/3) sin(−5π/6) = 5(
3/4
z = r cos(φ) = 5 cos(π/3) = 5(1/2) = 5/2
so (x, y, z) = (−15/4,−5
√
3/4, 5/2)
(c) x = r sin(φ) cos(θ) = 2 sin(π) cos(−π/6) = 2(0)(
√
3/2) = 0
y = r sin(φ) sin(θ) = 2 sin(π) sin(−π/6) = 2(0)(−1/2) = 0
z = r cos(φ) = 2 cos(π) = 2(−1) = −2
so (x, y, z) = (0, 0,−2)
(d) x = r sin(φ) cos(θ) = 8 sin(π/6) cos(9π/4) = 8(1/2)(
√
√
2/2) = 2
2
√
2/2) = 2
√
2
y = r sin(φ) sin(θ) = 8 sin(π/6) sin(9π/4) = 8(1/2)(
√
√
z = r cos(φ) = 8 cos(π/6) = 8(
3/2) = 4
3
√
√
√
3)
9. (a1) (4, π/3, 3π/4) =⇒ (4, 4π/3, π/4) =⇒ (4,−2π/3, π/4)
(a2) x = r cos p sin h = 4 cos(π/4) sin(−2π/3) = 4(
so (x, y, z) = (2
2, 2
2, 4
√
2/2)(−
√
3/2) = −
√
6
√
2/2) = −2
√
2
y = −r sin p = −4 sin(π/4) = −4(
√
2/2)(−1/2) = −
√
z = r cos p cos h = 4 cos(π/4) cos(−2π/3) = 4(
2
so (x, y, z) = (−
√
√
2,−
√
6,−2
2)
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