Example 5. Find spherical coordinates of point determined by Cartesian coordinates M=(6,0,8).

Solution. Spherical coordinate system is determined by

( S = O , p 1 = x , p 2 = z )

Spherical coordinates of point M are in the form M=(ρ,ϕ,ζ), where

ρ = x 2 + y 2 + z 2 = ( 6 ) 2 + 0 + ( 8 ) 2 = 100 = 10
ϕ = arccos x x 2 + y 2 = arccos 6 ( 6 ) 2 = arccos ( 1 ) = π
ζ = arccos x ρ = arccos 6 10 = π 5

and

M = ( 10, π , π 5 )
ex5
Figure 1: Spherical coordinates of point M