Year 9 Mathematics · AC9M9SP01

Why Sine, Cosine and Tangent Are Constant Ratios

Right triangles with the same acute angle are similar, so corresponding side ratios are constant. This similarity is the reason sine, cosine and tangent work

Ready to project and teach

Learning goalsSay it simply

Right triangles with the same acute angle are similar, so corresponding side ratios are constant. This similarity is the reason sine, cosine and tangent work.

By the end of this lesson, you should be able to:

  • use similarity to explain constant trigonometric ratios
  • identify opposite, adjacent and hypotenuse relative to an angle
  • connect side ratios to sine, cosine and tangent
  • use the ratio idea to justify right-triangle calculations
Key conceptTeach from the board

Same angle creates similar right triangles

Any two right triangles sharing the same acute angle have two equal angles, so they are similar by AA.

Similarity preserves corresponding side ratios

Although side lengths scale, opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent remain constant for the same angle.

Those constants are trig ratios

sinθ=opp/hyp, cosθ=adj/hyp and tanθ=opp/adj. The ratio depends on θ, not overall triangle size.

The reference angle controls the labels

Opposite and adjacent swap when the reference angle changes; the hypotenuse does not.

Worked examplesWe do

Worked examples

AC9M9SP01 - Why Sine, Cosine and Tangent Are Constant Ratios
Example 1

Scaled triangles A 3-4-5 triangle and 6-8-10 triangle have equal corresponding ratios because one is a scale copy.

Example 2

Sine constancy For the same θ, 3/5=6/10, so opposite/hypotenuse is unchanged.

Example 3

Change reference angle In one triangle, a side can be opposite one acute angle but adjacent to the other.

Example 4

Compare side ratios in a 3-4-5 and 6-8-10 triangle.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M9SP01: recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity

Victoria: VC2M9SP01 — Level 9 Space. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Pythagoras and trigonometry; Path — Further trigonometry. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9SP01VC2M9SP01 — Level 9 SpaceStage 5 Core — Pythagoras and trigonometry; Path — Further trigonometry
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
Why are trig ratios constant?
Right triangles with the same acute angle are similar, so corresponding side ratios are equal.
What never changes label?
The hypotenuse is always opposite the right angle.
What determines opposite and adjacent?
The chosen reference angle.
Practice and reviewReady for practice
  • Trig ratios change when a triangle is enlarged: Corresponding sides scale by the same factor, so ratios stay constant.
  • Opposite is always the vertical side: Opposite is defined relative to the chosen angle.
  • Hypotenuse depends on the angle: It is always opposite the right angle.
  • SOH-CAH-TOA is an unexplained rule: It follows from similarity and constant ratios.
  1. [4 marks] Prove two right triangles with the same acute angle are similar and identify equal side ratios.
  2. [5 marks] Use two scaled triangles to demonstrate numerical constancy of sine, cosine and tangent.
  3. [6 marks] Explain why a trig ratio can estimate an inaccessible height regardless of the measuring triangle’s size.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

Curriculum alignmentStart here

Right triangles with the same acute angle are similar, so corresponding side ratios are constant. This similarity is the reason sine, cosine and tangent work.

By the end of this lesson, you should be able to:

  • use similarity to explain constant trigonometric ratios
  • identify opposite, adjacent and hypotenuse relative to an angle
  • connect side ratios to sine, cosine and tangent
  • use the ratio idea to justify right-triangle calculations
Related resourcesKeep teaching
Teach & ExplainTeaching slides and samples

Teach this topic step by step

Explore optional teaching slide packs for classroom lessons and explanations at home.

Browse Teach & Explain · Browse Print & Go

Teachers: follow SkillrHub on TPT, then email us to request a free sample before buying. Include the year, subject and topic or curriculum code.

Request a free sample

After trying the sample, honest feedback is welcome. A TPT review is optional, where available, and does not need to be positive.