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)
Answer: (5.745, 0.8, 0.245)
