Abstract

This study concerns the portion of trigonometry which moves from right triangles to the coordinate plane, and then establishes sine and cosine as functions. The goal of the study was to explore students' understanding. A content framework was developed, and then applied in a case study of a group of students at the end of their work with this topic. The results were used to create a model of students' understanding of sine and cosine and also to refine the content framework. As part of the framework, a set of foundation concepts that underlie the subject of coordinate trigonometry was set forth. These ideas are called here the Trigonometric Connection.

The case study involved 120 honors students. The major data sources were a written test administered to the group and follow-up interviews with seven students. Both quantitative and qualitative methods were used to analyze the test scores. The qualitative analysis of the interviews provided insight into the thinking of the students as they answered the test questions. The study revealed that many students had an incomplete or fragmented understanding of the three major ways to view sine and cosine: as coordinates of a point on the unit circle, as the horizontal and vertical distances that are the graphical entailments of those coordinates, and as ratios of sides of a reference triangle. In addition, several cognitive obstacles were identified. Some involve issues specific to the study of trigonometry. These included a fragile conception of rotation angle and unit, and a failure to connect a rotation on the unit circle to a point on the graph of the cosine or sine function. Other obstacles relate to more fundamental topics. One obstacle is an inability to relate the coordinates of a point on a graph to the horizontal and vertical segments that connect the point to the axes. Another difficulty relates to the nature of sine and cosine as both ratios and numbers.