In Calculus, by a function, $\;f$, we mean the rule which assigns to each element $x \in D$ (domain) an element $y \in \mathbb R$ (range). Where $x$ and $y$ are real numbers
According to the definition above, the trig function should be regarded as single-valued. But can trig functions operate on domains consisting of angles instead of real numbers? If not, how can we allow trig functions operate on domains consisting of real numbers? provided that we can't disregard traditional trigonometry because it exists beforehand. Whats unambiguous with traditional trigonometry that made mathematicians rethink and use real numbers rather than angles?