### Instruction: In spherical coordinates, the point $\left( 3, \frac{2 \pi}{7}, \frac{8 \pi}{5} \right)$ is equivalent to what other point, in the standard spherical coordinate representation?  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 recall that spherical coordinates are related to Cartesian coordinates by the formulas $x = \rho \sin \phi \cos \theta,$ $y = \rho \sin \phi \sin \theta,$ and $z = \rho \cos \phi.$
To find the equivalent point in the standard representation, I need to adjust the values of $\theta$ and $\phi$ so that they are in the required ranges.
I notice that the value of $\theta$ is already in the range $[0,2 \pi),$ so I don't need to change it.
The value of $\phi,$ however, is not in the range $[0,\pi].$
To fix this, I can use the fact that $\sin \phi$ is periodic with period $2 \pi$ and symmetric about $\pi.$
This means that $\sin \phi = \sin (\phi + 2 \pi) = \sin (2 \pi - \phi).$
So, I can replace $\phi$ with either $\phi + 2 \pi$ or $2 \pi - \phi,$ depending on which one is in the range $[0,\pi].$
In this case, $\phi + 2 \pi > \pi,$ so I choose $2 \pi - \phi.$
This gives me the equivalent point $\left( 3, \frac{2 \pi}{7}, 2 \pi - \frac{8 \pi}{5} \right).$
I simplify this by writing $2 \pi - \frac{8 \pi}{5} = \frac{2 \pi}{5}.$