Answer:
(x ,y ,z) = (√3 ,3 ,2) (rectangular coordinates)
(ρ, θ, φ) = (4 , π/3, π/3) (spherical coordinates)
Step-by-step explanation:
if the cylindrical coordinates are
( r , θ , z) = (2√3, π/3, 2)
then assuming that the θ angle is the one in the x-y plane relative to the x axis :
x= r*cos θ = 2√3*cos (π/3) = 2√3* (1/2)= √3
y= r*sin θ = 2√3*sin (π/3) = 2√3* (√3/2) = 3
z = z =2
then (x ,y ,z) = (√3 ,3 ,2) (rectangular coordinates)
denoting φ as the angle relative to the z axis and ρ as the distance to the origin
ρ = √(x²+y²+z²) = √(3+9+4) = 4
z = ρ*cos φ → φ = cos⁻¹ (z/ρ) =cos⁻¹ (1/2) = π/3
then (ρ, θ, φ) = (4 , π/3, π/3) (spherical coordinates)
