Spherical Coordinates — Question 5

Question 5 – Solution

Strategy. Substitute $z=\rho\cos\phi$ and $x^2+y^2=\rho^2\sin^2\phi$, treating $\rho=0$ separately.

Step 1: Convert For $\rho>0$, $$\rho^2\cos^2\phi=3\rho^2\sin^2\phi\quad\Longrightarrow\quad \tan^2\phi=\frac 13.$$ On $0\le\phi\le\pi$, the solutions are $$\boxed{\phi=\frac\pi 6\quad\text{or}\quad\phi=\frac{5\pi}{6}}.$$

Step 2: Geometry The first value is the upper nappe, $z\ge 0$; the second is the lower nappe, $z\le 0$. Each makes an acute angle $\pi/6$ with its adjacent half of the $z$-axis.

Step 3: Origin At $\rho=0$, every angle represents the origin, which satisfies the Cartesian equation. Therefore the two nappes meet at the included vertex $\boxed{(0,0,0)}$.