How do I write this in spherical coordinates?

Question

$$z=\sqrt{2x^2+2y^2}$$ I know that this is $\tan\phi=\frac{1}{\sqrt{2}}$ but the arctan of that number is 0.6154797, so I'm confused with what to put as the spherical coordinate.

Answer 1

Spherical co-ordinates are described by $$x=r\sin\theta\cos\phi$$ $$y=r\sin\theta\sin\phi$$ $$z=r\cos\theta$$ You have:$$2(x^2+y^2)=z^2\implies2(r^2\sin^2\theta(\cos^2\phi+\sin^2\phi))=r^2\cos^2\theta$$

$$\implies2r^2\sin^2\theta=r^2\cos^2\theta\implies\sin^2\theta=\frac13,\cos^2\theta=\frac23\implies\tan^2\theta=\frac12$$ as you got.