Question

In Exercises \(41-46,\) convert the point from cylindrical coordinates to spherical coordinates. $$ (4, \pi / 2,3) $$

Short answer

The spherical coordinates for the given point in cylindrical coordinates are (5, \(arctan(4/3)\), \(π/2\)).

Step-by-step solution

Identify cylindrical coordinates

The cylindrical coordinates given are (r, θ, z) = (4, \(π/2\), 3).

Convert radius to spherical coordinate ρ

Using the formula \( ρ = √(r^2 + z^2) \), replace r with 4 and z with 3. Therefore, \( ρ = √((4)^2 + (3)^2) = √(16+9) = √25 = 5 \).

Find the φ coordinate

Using the formula \( φ = arctan(r/z) \), replace r with 4 and z with 3. So, \( φ = arctan(4/3) \). Note that we should express this in the usual range for spherical coordinates, which is \( φ ∈ [0, π] \).

Identify θ coordinate

In spherical coordinates, the θ coordinate remains unchanged from the cylindrical coordinate system. Hence, θ remains \(π/2\). \nSo, the point (4, \(π/2\), 3) in cylindrical coordinates corresponds to (5, \(arctan(4/3)\), \(π/2\)) in spherical coordinates.