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• We present the results of presenting an instructional module connecting the teaching of real analysis to the teaching of secondary mathematics. • We focus on the relationship between continuity, injectivity, and invertibility and the use of the arcsine function. • Prospective teachers who completed this module were better prepared to teach inverse trigonometric functions. Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers' instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter. [ABSTRACT FROM AUTHOR]