This research explores two methods through which ten preservice math teachers develop an understanding of trigonometric values. Using the unit circle, preservice math teachers engage in knowledge-building activities such as paper folding and GeoGebra application. Grounded in Altman and Kidron's 2016 didactical design research, this study examines the cognitive processes ten preservice math teachers undergo during knowledge acquisition procedure. Employing the dynamically nested epistemic action model for abstraction, we analyze how different tasks facilitate ten preservice math teachers' comprehension of unit circle representations for trigonometric expressions and their associated values. Furthermore, we apply the abstraction in context framework to observe how ten preservice math teachers' knowledge progresses from traditional 'triangle' trigonometry to 'circle' trigonometry, aiding in determining trigonometric values. [This paper is published in: W-C Yang, D. Meade, W. K. Ho (Eds.), "Proceedings of the 29th Asian Technology Conference in Mathematics (ATCM 2024)," (pp. 218-228). Yogyakarta, Indonesia.]