Students explore how the angles and side length properties of similar right-angled triangles may be applied in measurement situations to calculate unknown side lengths and angles.

Teacher notes

  • There is a progression from the informal ratio techniques used in ancient cultures to the more formal methods of right-angled triangle trigonometry.
  • A step-by-step approach with scaffolding and informative feedback leads students through establishing similarity, identifying corresponding sides, establishing equivalent ratios and finding unknown lengths.
  • Introduces students to the formal trigonometric ratios sine, cosine and tangent and their use in triangle measurement contexts.
  • Demonstrates and maintains the fundamental underpinning links between similarity and triangle trigonometry.

Learning objects

trigonometry picture

Trigonometry: measuring with triangles
Students see how the early Greeks and Egyptians used the properties of similar right-angle triangles to solve measurement problems. They then identify and record the relevant ratios in two similar right-angle triangles to find the unknown height of one of them (represented as the height of a column).
Students then use ratio in three pairs of similar isosceles right-angle triangles to find the unknown value on one triangle.

trigonometry picture.

Trigonometry: similar triangles
Students manipulate and identify the properties of similar right-angle triangles.

trigonometry picture.

Trigonometry: sine
Students use a 'unit circle' tool to complete a table of values concerning sine.

trigonometry picture.

Trigonometry: cosine
Students use a 'unit circle' tool to complete a table of values concerning cosine.

trigonometry picture.

Trigonometry: tangent
Students use a 'unit circle' tool to complete a table of values concerning tan.

trigonometry picture.

Trigonometry: using sine
Students use sine and the properties of the right-angle triangles to solve measurement problems.

trigonometry picture.

Trigonometry: using cosine
Students use cosine and the properties of the right-angle triangles to solve measurement problems.

trigonometry picture.

Trigonometry: using tan
Students use tan and the properties of the right-angle triangles to solve measurement problems.

trigonometry picture.

Trigonometry: finding the hypotenuse
Students use sine to find the hypotenuse.

trigonometry picture.

Trigonometry: finding the angles
Students use tri ratios to find missing angles.