The (say) real sine function is defined such that its domain and codomain are (subsets of) the reals. The reals are usually characterized as the complete ordered field. I have never come across units that--taken alone--satisfy the axioms of a complete ordered field, and having several units introduces problems such as how we would impose a meaningful order. So a sine function over unit-ed quantities is sufficiently non-obvious as to require a clarification of what would be meant by sin($1). For example--switching over now to logarithms--if we treat $1 as the real multiplicative identity (i.e. the real number, unity) unit-multiplied by the unit $, and extrapolate one of the fundamental properties of logarithms--that log(ab)=loga+logb, we find that log($1)=log($)+log(1)=log($) (assuming we keep that log(1)=0). How are we to interpret log($)? Moreover, log($^2)=2log($). So if I log the square of a dollar, I obtain twice the log of a dollar. How are we to interpret this in the above context of utility? Or an example from trigonometric functions: One characterization of the cosine and sine stipulates that cos^2+sin^2=1, so we would have that cos^2($1)+sin^2($1)=1. If this is the real unity, does this mean that the cosine function on dollars outputs a real number? Or if the RHS is $1, does this mean that the cosine function on dollars outputs a dollar^(1/2) value? Then consider that double, triple, etc. angles in the standard cosine function can be written as polynomials in the single-angle cosine. How would this translate?

So this is a case where the 'burden of meaningfulness' lies with proposing a meaningful interpretation (which now seems rather difficult), even though at first it seems obvious that there is a single reasonable way forward. The context of the functions needs to be considered; the sine function originated with plane geometry and was extended to the reals and then the complex numbers. Each of these was motivated by an (analytic) continuation into a bigger 'domain' that fit perfectly with existing understanding of that bigger domain; this doesn't seem to be the case here.