### Instruction: A point has rectangular coordinates $(8,-3,-1)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi).$.### Response: I know that the relationship between rectangular and spherical coordinates is given by the formulas $x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, z = \rho \cos \phi.$
So, to find the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi),$ I just need to plug in these values into the formulas and simplify.
I get $x = \rho \sin \phi \cos (-\theta) = \rho \sin \phi \cos \theta,$ since cosine is an even function.
Similarly, $y = \rho \sin \phi \sin (-\theta) = -\rho \sin \phi \sin \theta,$ since sine is an odd function.
And $z = \rho \cos \phi,$ which is unchanged.
So, the rectangular coordinates of the point with spherical coordinates $(\rho, -\theta, \phi)$ are the same as the original point, except the $y$-coordinate is negated.
Therefore, the answer is $(8, 3, -1).$