The unit circle approach to the construction of the sine and cosine functions and their inverses: An application of APOS theory

• Proposes a genetic decomposition of the sine, cosine, and their inverse functions.
• Genetic decomposition is based on a small number of “basic mental constructions”.
• Genetic decomposition is useful to describe student behavior.
• Includes inverse trigonometric functions, a topic neglected in the literature.
• Has practical applications to improve student understanding of these functions.

We use Action-Process-Object-Schema (APOS) Theory to analyze the mental constructions made by students in developing a unit circle approach to the sine, cosine, and their corresponding inverse trigonometric functions. Student understanding of the inverse trigonometric functions has not received much attention in the mathematics education research literature. We conjectured a small number of mental constructions, (genetic decomposition) which seem to play a key role in student understanding of these functions. To test and refine the conjecture we held semi-structured interviews with eleven students who had just completed a traditional college trigonometry course. A detailed analysis of the interviews shows that the conjecture is useful in describing student behavior in problem solving situations. Results suggest that students having a process conception of the conjectured mental constructions can perform better in problem solving activities. We report on some observed student mental constructions which were unexpected and can help improve our genetic decomposition.