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Section 2.4 Spherical Coordinates

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The spherical coordinates (ρ,θ,ϕ) of a point (x,y,z) are defined as follows:
  • From cartesian to spherical.
    ρ=x2+y2+z2,θ=tan−1⁡(yx), and ϕ=cos−1⁡(zx2+y2+z2)
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    We follow the same conditions on θ as with the cylindrical coordinates.
  • From spherical to cartesian.
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    x=ρsin⁡ϕcos⁡θ,y=ρsin⁡ϕsin⁡θ, and z=ρcos⁡ϕ.
    where
    ρ≥0,0≤θ≤2π,0≤ϕ≤π.
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What are the cylindrical coordinates of the point whose spherical coordinates are
(2, 5, 4π6) ?
r =
θ =
z=
Answer 1.
1.73205080756888
Answer 2.
5
Answer 3.
−1
What are the spherical coordinates of the point whose rectangular coordinates are
(3, 3, −3) ?
ρ =
θ =
ϕ =
Answer 1.
5.19615242270663
Answer 2.
0.785398163397448
Answer 3.
2.18627603546528
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