The equation given, x² + y² + z² = 16, represents a sphere centered at the origin with a radius of 4. To convert this equation into spherical coordinates, we start by recalling the relationships between Cartesian and spherical coordinates:
- x = ρ sin(φ) cos(θ)
- y = ρ sin(φ) sin(θ)
- z = ρ cos(φ)
Here, ρ represents the radius (the distance from the origin), φ is the polar angle (the angle from the positive z-axis), and θ is the azimuthal angle (the angle in the xy-plane from the positive x-axis).
Now, we can substitute these relationships into the original equation:
Substituting:
(ρ sin(φ) cos(θ))² + (ρ sin(φ) sin(θ))² + (ρ cos(φ))² = 16
This simplifies as follows:
ρ² (sin²(φ) cos²(θ) + sin²(φ) sin²(θ) + cos²(φ)) = 16
Factor out ρ²:
ρ² (sin²(φ) (cos²(θ) + sin²(θ)) + cos²(φ)) = 16
Using the identity (cos²(θ) + sin²(θ) = 1), we have:
ρ² (sin²(φ) + cos²(φ)) = 16
Since (sin²(φ) + cos²(φ) = 1), we can further simplify to:
ρ² = 16
Finally, by taking the square root of both sides, we find:
ρ = 4
This means that in spherical coordinates, the equation of the sphere is simply:
ρ = 4
This indicates all points that are 4 units away from the origin in three-dimensional space.