This article begins by defining two sets of related constants: The first set is used to define a nonstandard cubic equation. The second set is used to define equations that constrain a pair of rotation matrices. Curiously, these sets each contain an angle, each defined much like the other, where these angles separately play a central role in identifying solutions to their respective, very different equations. In this way these similarly-defined angles provide good evidence of a non-coincidental relationship between the nonstandard cubic equation and the pair of matrices. Moreover, a special case of this nonstandard cubic neatly combines with these matrices to produce values that map over to multiple physical constants (the fine structure constant reciprocal, the Weinberg angle, and the quark and lepton mixing angles), providing further evidence of a non-coincidental relationship.