Convert the following into spherical coordinates: x^2+y^2+Z^2=36
To convert the equation x^2 + y^2 + z^2 = 36 into spherical coordinates, we need to express x, y, and z in terms of the spherical coordinates: ρ, θ, and φ.
In spherical coordinates, ρ represents the distance from the origin to the point, θ represents the angle from the positive x-axis to the projection of the point on the xy-plane, and φ represents the angle between the positive z-axis and the line connecting the origin to the point.
We can express x, y, and z in terms of the spherical coordinates as follows:
x = ρ*sin(φ)*cos(θ)
y = ρ*sin(φ)*sin(θ)
z = ρ*cos(φ)
Substituting these expressions into the equation x^2 + y^2 + z^2 = 36, we get:
(ρ*sin(φ)*cos(θ))^2 + (ρ*sin(φ)*sin(θ))^2 + (ρ*cos(φ))^2 = 36
Simplifying this equation, we obtain:
ρ^2*sin^2(φ)*cos^2(θ) + ρ^2*sin^2(φ)*sin^2(θ) + ρ^2*cos^2(φ) = 36
ρ^2 * (sin^2(φ)*cos^2(θ) + sin^2(φ)*sin^2(θ) + cos^2(φ)) = 36
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can simplify further:
ρ^2 * (sin^2(φ)*(cos^2(θ) + sin^2(θ)) + cos^2(φ)) = 36
Since cos^2(θ) + sin^2(θ) = 1, we can write the equation as:
ρ^2 * (sin^2(φ) + cos^2(φ)) = 36
ρ^2 * 1 = 36
ρ^2 = 36
Taking the square root of both sides, we get:
ρ = 6
Therefore, in spherical coordinates, the equation x^2 + y^2 + z^2 = 36 is equivalent to ρ = 6, where ρ represents the distance from the origin to the point.