<![CDATA[Many university students experience difficulties connecting graphical, symbolic, and analytical representations when solving extended-angle trigonometric problems, resulting in fragmented conceptual understanding. Previous studies have mainly examined misconceptions or instructional media separately, while limited research has explored how students transition between multiple mathematical representations. This study employed an explanatory sequential mixed-methods approach integrating quantitative and qualitative analyses to investigate students’ representational thinking patterns in extended-angle trigonometry learning. Students’ mean scores increased significantly from 54.30 to 75.80 (t = -8.92, p < 0.001), indicating substantial improvement in conceptual understanding after the instructional intervention. The findings revealed diverse thinking patterns, with most students relying on visual representations, while others preferred symbolic procedures. Students demonstrating analytical flexibility showed deeper representational coordination and conceptual comprehension, enabling smoother transitions among visualization, symbolic manipulation, and analytical reasoning. Principal Component Analysis (PCA) was employed for dimensionality reduction and visualization of students’ cognitive tendencies, while thematic analysis of interviews and think-aloud protocols provided deeper insights into students’ problem-solving processes. The study contributes a cognitive-transition framework explaining how students coordinate visualization, symbolic manipulation, and analytical reasoning in trigonometric problem solving and provides implications for designing adaptive instructional strategies that support representational flexibility and conceptual understanding.]]>
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