Question

Find an equation in rectangular coordinates for the spherical equation?

Answer 1

`\(x = r \cdot \sin(\theta) \cdot \cos(\phi)\), \(y = r \cdot \sin(\theta) \cdot \sin(\phi)\), \(z = r \cdot \cos(\theta)\)`This conversion formula allows for the translation between spherical coordinates (r, θ, φ) and Cartesian coordinates (x, y, z),

Step-by-step explanation:

The equation in rectangular coordinates for the spherical equation is derived through trigonometric relationships. Converting from spherical to rectangular coordinates involves using the relationships between the spherical coordinates (r, θ, φ) and the Cartesian coordinates (x, y, z). For x-coordinate, it's obtained by multiplying r (the radial distance) by the sine of θ (the inclination angle) and the cosine of φ (the azimuthal angle).

The y-coordinate is determined by multiplying r by the sine of θ and the sine of φ. Lastly, the z-coordinate is derived by taking r and multiplying it by the cosine of θ. These conversions facilitate translating the spherical coordinates into their respective Cartesian counterparts.

This conversion formula allows for the translation between spherical coordinates (r, θ, φ) and Cartesian coordinates (x, y, z), providing a clear understanding of a point's position in 3D space using different coordinate systems.