Question

Derive an equation for the vertical coordinate y of the sphere.

a) y = r sin(θ)
b) y = r cos(θ)
c) y = r tan(θ)
d) y = r sec(θ)

Answer 1

The correct equation for the vertical coordinate y of a sphere in the spherical coordinate system is y = r sin(θ), based on the spherical to Cartesian coordinate transformations.

Step-by-step explanation:

The student's question pertains to deriving the equation for the vertical coordinate y of a sphere in the spherical coordinate system. In this system, the vertical coordinate y is obtained by projecting the position vector of a point onto the vertical axis.

Based on the information that the spherical coordinates are related to the Cartesian coordinates by the equations x = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ, we can determine which of the provided equations is correct for the vertical coordinate y.

The correct equation for the vertical coordinate y in terms of the spherical coordinates is y = r sin(θ), which corresponds to option (a). The function sine is used because y relates to the projection on the y-axis, and the angle θ is the polar angle, which implies a sine relationship for the vertical component in polar and spherical coordinates.