Question

The cylindrical coordinates of a point are (2√3, π/3, 2). Find the rectangular and spherical coordinates of the point.

Answer 1

(x ,y ,z) = (√3 ,3 ,2) (rectangular coordinates)

(ρ, θ, φ) = (4 , π/3, π/3) (spherical coordinates)

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)