What are the spherical coordinates of the point whose rectangular coordinates are (4,1,4)?

Refer to the figure shown below.

The transformation between the rectangular (cartesian) coordinate system (x, y, z) and the orthogonal spherical coordinate system (r, θ, φ) s given by these formulas:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ

Also,
r = √(x² + y² + z²)
θ = cos⁻¹ (z/r)
φ = tan⁻¹(y/x)

Given: (x,y,z) = (4,1,4).
Therefore
r = √(16+1+16) = 5.745
θ = cos⁻¹(4/5.745) = 45.9° = 0.8 rad
φ = tan⁻¹(1/4) = 14.04° = 0.245 rad

Spherical coordinates are usually given in radians.
The spherical coordinates are (5.745, 0.8, 0.245)

(5.745, 0.8, 0.245)

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