### 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: To convert from spherical coordinates to standard spherical coordinates, I need to adjust the angles $\theta$ and $\phi$ so that they fit the ranges given in the answer format.
The angle $\theta$ is the azimuthal angle, which measures the angle from the positive $x$-axis in the $xy$-plane.
The angle $\phi$ is the polar angle, which measures the angle from the positive $z$-axis.
The radius $\rho$ is the distance from the origin to the point, and it does not change under the conversion.
So, the radius of the point is still $3$.
To adjust the angle $\theta$, I need to add or subtract multiples of $2 \pi$ until it lies between $0$ and $2 \pi$.
The given angle $\theta$ is $\frac{2 \pi}{7}$, which is already in the range, so I do not need to change it.
To adjust the angle $\phi$, I need to reflect it across the $xy$-plane if it is greater than $\pi$, and then add or subtract multiples of $\pi$ until it lies between $0$ and $\pi$.
The given angle $\phi$ is $\frac{8 \pi}{5}$, which is greater than $\pi$, so I need to reflect it across the $xy$-plane.
This means subtracting it from $2 \pi$, which gives $2 \pi - \frac{8 \pi}{5} = \frac{2 \pi}{5}$.
This angle is already in the range, so I do not need to change it further.
Therefore, the standard spherical coordinates of the point are $\left( 3, \frac{2 \pi}{7}, \frac{2 \pi}{5} \right)$.