Final answer:
To convert from rectangular coordinates to spherical coordinates, calculate the radial distance r, the polar angle θ, and the azimuthal angle φ, using the respective formulas r = √(x² + y² + z²), θ = arccos(z / r), and φ = atan2(y, x).
Step-by-step explanation:
Converting rectangular coordinates to spherical coordinates involves the calculation of three values: the radial distance (r), the polar angle (θ), often referred to as the inclination or colatitude, and the azimuthal angle (φ). The radial distance is the direct distance from the origin to the point in question. The polar angle is the angle made with the positive z-axis, and the azimuthal angle is the angle made with the positive x-axis in the xy-plane.
Starting with the given rectangular coordinates (x, y, z), we can find the radial distance using the formula r = √(x² + y² + z²). Next, the polar angle is obtained using θ = arccos(z / r), and the azimuthal angle is calculated with φ = atan2(y, x).
These formulas allow for the transition from a Cartesian grid, like what we would see on a city map, to a spherical system similar to latitude and longitude on a globe.