The transition from calculus to analysis presents well known challenges to many students. In line with Chevallard's “paradigm of questioning the world”, we consider that part of the difficulty is a failure to question (explore) what the formal theory of analysis means for “familiar objects” from calculus. As a significant case, we consider the familiar notions of “angle”, cosine and sine. We first examine some “advanced” viewpoints on these objects coming from analysis, then the viewpoints an advanced student could produce on these notions, based on knowledge from an analysis course in which resources for a rigorous account were indeed available.