Answer to Question #346325 in Analytic Geometry for James

Question

Identify the surface of the ρ = sinΦsinθ by converting them into equations in the Cartesian form. Show the complete solutions.

Answer 1

$$x=\rho\sin\phi\cos\theta, y=\rho\sin\phi\sin\theta, z=\rho\cos\phi$$

$$\rho^2=x^2+y^2+z^2$$

Given

$$\rho=\sin\phi\sin\theta$$

Then

$$\rho^2=\rho\sin\phi\sin\theta$$

Substitute

$$x^2+y^2+z^2=y$$

$$x^2+(y-\dfrac{1}{2})^2+z^2=(\dfrac{1}{2})^2$$

The surface is a sphere.