### Instruction: A point has rectangular coordinates $(x,y,z)$ and spherical coordinates $\left(2, \frac{8 \pi}{7}, \frac{2 \pi}{9} \right).$  Find the spherical coordinates of the point with rectangular coordinates $(x,y,-z).$  Enter your answer in the form $(\rho,\theta,\phi),$ where $\rho > 0,$ $0 \le \theta < 2 \pi,$ and $0 \le \phi \le \pi.$.### Response: I know that the conversion formulas between rectangular and spherical coordinates are:

\begin{align*}
x &= \rho \sin \phi \cos \theta \\
y &= \rho \sin \phi \sin \theta \\
z &= \rho \cos \phi
\end{align*}
I also know that the point $(x,y,-z)$ is the reflection of $(x,y,z)$ across the $xy$-plane, so the distance $\rho$ from the origin and the angle $\theta$ in the $xy$-plane are the same for both points.
Only the angle $\phi$ from the positive $z$-axis changes.
Therefore, I can write the spherical coordinates of $(x,y,-z)$ as $(\rho,\theta,\pi - \phi)$, where $\rho$ and $\theta$ are the same as for $(x,y,z)$.