Pre-knowledge

The Concept of Similarity: Students need to have a command of the concept of similarity and have worked through the conditions for similar triangles.

Knowledge of Right-Angled Triangle Properties: Students identify the sides of a right-angled triangle, hypotenuse ($H$), adjacent ($A$), and opposite ($O$), relative to a given reference angle ($\theta$) and use these labels to determine corresponding sides and angles. When studying Trigonometry A, they build upon their understanding of the relationships between the corresponding sides of similar right-angled triangles to investigate trigonometric ratios. This understanding forms the foundation of trigonometric ratios and will further support students in exploring the concept of the unit circle in later learning. (Education Apps, n.d.)

Pythagoras’ Theorem Application: Students must be able to apply Pythagoras’ theorem in a right-angled triangle to find unknown side lengths.

Potential Difficulties

According to Williams (2019), when learning trigonometry, students often rely heavily on procedural knowledge, such as memorisation, while lacking a conceptual understanding of the fundamental trigonometric ratios as comparisons between sides in a right-angled triangle. As a result, they are usually able to solve familiar, procedural problems (for example, finding $\sin \theta$ when all side lengths are given) but tend to struggle with tasks that require deeper conceptual reasoning.

Lack of Ratio Concept: Students struggle to view the sine function as a ratio of the length of two sides ($\frac{\text{opposite}}{\text{hypotenuse}}$). They were unable to find or estimate the sine ratio when no specific side lengths were given.

Inability to Generalise: Students can use procedures to find the ratios of common angles but cannot expand their understanding to estimate the ratios for non-typical angles (e.g., 20°), which is necessary for interpreting and solving practical problems.

Students show a lack of conceptual understanding when they believe they must be given numerical side lengths or angle measurements to find or estimate a trigonometric ratio.

Common Misconceptions

Confused About Corresponding Sides: Many students mistakenly assume that corresponding sides are the ones that look alike, even if the shapes are rotated differently. Aligning the shapes in the same direction helps them see which sides correspond. Teachers should remind students to compare the entire figures, not just individual parts, when deciding if shapes are similar. This idea connects to their earlier learning about similar figures and right-angled triangles.

Linear Scaling Error: Students incorrectly assume that trigonometric functions are linearly proportional to the angle.

Mixing up $\boldsymbol{\sin}$ and $\boldsymbol{\cos}$: Students make errors in identifying the trigonometric ratios, such as incorrectly identifying the x-coordinate as the sine ratio when recalling memorised values. This reveals a confusion in which side corresponds to which ratio.

Mixing up Reciprocal and Inverse Function Confusion: Students frequently confuse reciprocal trigonometric functions ($\csc \theta = \frac{1}{\sin \theta}$) with inverse trigonometric functions ($\sin^{-1} x$), particularly when attempting to use a calculator to find an unknown angle.

References

Education Apps. (n.d.). Introducing trigonometric ratios in right-angled triangles. Retrieved October 11, 2025, from https://arc.educationapps.vic.gov.au/learning/sites/mcc/VCMMG319/Introducing-trigonometric-ratios-in-right-angled-triangles

Williams, C. (2019, December 13). Understandings and misunderstandings of trigonometry. Georgia College & State University. https://www.gcsu.edu/sites/files/page-assets/node-808/attachments/williams.pdf