I certainly take your point. I should clarify  my question relates specifically to the "complement" of an complex angle of refraction, and its physical interpretation in the context of total internal reflection.
In this case, the complement I mean to obtain is the one with physical significance in the context of Snell's law (or the equivalent Fresnel expressions) when the incident angle exceeds the critical angle. If Snell's law was defined with angles relative to the interface, at TIR this refracted angle would be the complex number I am looking for.
I still confused about whether simply taking [itex]Pi/2  θ_{t}[/itex] makes physical sense in this context.
