### 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 the spherical coordinates of a point are $(\rho,\theta,\phi),$ where $\rho$ is the distance from the origin, $\theta$ is the angle in the $xy$-plane, and $\phi$ is the angle from the positive $z$-axis.
I notice that the given point has $\theta = \frac{2 \pi}{7},$ which is already in the standard range of $0 \le \theta < 2 \pi.$
So I don't need to change that coordinate.
I also notice that the given point has $\phi = \frac{8 \pi}{5},$ which is greater than $\pi.$
This means that the point is below the $xy$-plane, and I can find an equivalent point above the $xy$-plane by subtracting $\phi$ from $2 \pi.$
So I calculate the new value of $\phi$ as $2 \pi - \frac{8 \pi}{5} = \frac{2 \pi}{5}.$
Finally, I notice that the given point has $\rho = 3,$ which is positive.
So I don't need to change that coordinate either.
Therefore, the equivalent point in the standard spherical coordinate representation is $\left( 3, \frac{2 \pi}{7}, \frac{2 \pi}{5} \right).$