What is $R = 2\sin\theta \sin\phi$ (spherical coordinates) in cartesian coordinates?
Hello I'm trying to figure out how to turn $R = 2\sin\theta \sin\phi$ (spherical coordinates) in cartesian coordinates.
My attempt
$$\begin{align}
&R = 2\sin\theta \sin\phi \\
&Rr = 2r\sin\theta \sin\phi \\
&Rr = 2y \\
&R \sqrt{x^2+y^2} = 2y\\
&(x^2+y^2+z^2)\sqrt{x^2+y^2} = 2y
\end{align}$$
Can somebody tell me if I'm doing this right, because this looks wrong. If you can help me, please do
Thanks in advance
You have made a mistake in your identity for $y$. In general spherical coordinates, we have the following identities $$r=R\sin\phi$$ $$y=r\sin\theta=R\sin\phi\sin\theta$$ To apply these to the given equation, $R=2\sin\phi\sin\theta$, multiply both sides by $R$, $$R^2=2R\sin\phi\sin\theta$$ $$R^2=2y$$ $$x^2+y^2+z^2=2y$$ $$\boxed{x^2+(y-1)^2+z^2=1}$$