Question

In Exercises \(41-46,\) convert the point from cylindrical coordinates to spherical coordinates. $$ (4, \pi / 4,0) $$

Short answer

The spherical coordinates of the point (4, \(\pi / 4\), 0) are (4, 0, \(\pi / 4\))

Step-by-step solution

Identify the given values

The given cylindrical coordinates are (4, \(\pi / 4\), 0). Here, p=4, \(\theta=\pi / 4\), and z=0.

Calculate the value of r

Using \(r=\sqrt{p^2+z^2}\), let's plug the values of p and z into it. This results in \(r=\sqrt{4^2+0^2}=\sqrt{16}=4\).

Calculate the value of phi

Using \(\phi=\arctan(z/p)\), plug in the known values. Since z=0, we get \(\phi=0\).

The angle theta

The angle theta remains the same in both cylindrical and spherical coordinates. Hence, \(\theta=\pi / 4\).