Rethinking the Pedagogical Sequence of Trigonometric Formulae in Secondary Education : A Preliminary Assessment of an Alternative Teaching Approach in a Hong Kong Secondary

Abstract

The teaching of trigonometry occupies an important position in secondary mathematics education because it 
serves as a bridge between geometry, algebra, and advanced mathematical modelling. Despite its importance, 
many students encounter significant difficulties when solving non-right-angled triangles using the sine and 
cosine formulae. In Hong Kong secondary schools, the conventional textbook sequence typically introduces the 
sine formula through angle-side-angle cases, proceeds to the ambiguous side-side-angle case, and subsequently 
introduces the cosine formula through side-angle-side and side-side-side situations. While this sequence reflects 
mathematical logic and curriculum traditions, its pedagogical effectiveness has received relatively little scrutiny. 
This conceptual study re-examines the sequencing of trigonometric formulae from the perspective of classroom 
teaching and learning. Drawing upon reflective practice developed through secondary mathematics teaching in 
Hong Kong, the study argues that the conventional ordering may impose unnecessary cognitive demands on 
learners, particularly when the ambiguous case of the sine formula is introduced prematurely. An alternative 
pedagogical approach is proposed whereby students first encounter area formulae, followed by the sine rule as 
a tool for determining unknown lengths, while the cosine rule is employed strategically to reduce ambiguity and 
support exact algebraic solutions. 
The manuscript analyses the mathematical structure of solving triangles, common student misconceptions, 
examination errors, and the interaction between conceptual understanding and procedural fluency. Relevant 
theoretical perspectives, including cognitive load theory, constructivism, variation theory, and APOS theory, 
are employed to frame the discussion. Implications for curriculum design, textbook development, teacher 
education, assessment practices, and future research are also explored. 
Rather than presenting a large-scale empirical investigation, this study contributes a theoretically informed 
pedagogical framework intended to stimulate professional dialogue concerning the teaching of trigonometry in 
secondary schools. The analysis suggests that curriculum sequencing deserves greater attention and that 
alternative instructional pathways may better support students' conceptual understanding and problem-solving 
competence.