Convert the following into spherical
coordinates: X^2+y^2+Z^2 = 36.
To convert an equation from Cartesian coordinates to spherical coordinates, we need to express the x, y, and z terms in terms of the spherical coordinates variables: ρ (rho), θ (theta), and φ (phi).
In spherical coordinates, ρ represents the radial distance from the origin to the point, θ represents the polar angle measured from the positive z-axis, and φ represents the azimuthal angle measured from the positive x-axis.
To convert the equation X^2 + y^2 + Z^2 = 36 into spherical coordinates, we can substitute the Cartesian coordinates variables, x, y, and z, with the corresponding spherical coordinates formulas:
x = ρ * sin(φ) * cos(θ)
y = ρ * sin(φ) * sin(θ)
z = ρ * cos(φ)
Substituting these equations into the given equation, we have:
(ρ * sin(φ) * cos(θ))^2 + (ρ * sin(φ) * sin(θ))^2 + (ρ * cos(φ))^2 = 36
Simplifying this equation, we get:
ρ^2 * sin^2(φ) * cos^2(θ) + ρ^2 * sin^2(φ) * sin^2(θ) + ρ^2 * cos^2(φ) = 36
Rearranging the terms, we have:
ρ^2 * (sin^2(φ) * cos^2(θ) + sin^2(φ) * sin^2(θ) + cos^2(φ)) = 36
Combining the terms within the parentheses, we get:
ρ^2 * (sin^2(φ) * (cos^2(θ) + sin^2(θ)) + cos^2(φ)) = 36
Since cos^2(θ) + sin^2(θ) equals 1, we can further simplify the equation to:
ρ^2 * (sin^2(φ) + cos^2(φ)) = 36
Simplifying the trigonometric terms, we have:
ρ^2 = 36
Taking the square root of both sides, we find:
ρ = ±6
Therefore, the equation X^2 + y^2 + Z^2 = 36 in spherical coordinates is ρ = ±6.