Why Sine, Cosine and Tangent Are Constant Ratios — AC9M9SP01

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

Recall similar triangles, corresponding sides, ratios, right angles and basic right-triangle trigonometry vocabulary.

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.

• 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. Compare side ratios in a 3-4-5 and 6-8-10 triangle.
2. For a labelled right triangle, identify opposite, adjacent and hypotenuse for θ.
3. Explain using similarity why sinθ is the same in two scaled triangles.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Create two similar right triangles and compare all three trig ratios.
2. Explain why doubling every side leaves tanθ unchanged.
3. A side is opposite angle A. Can it be adjacent to angle B? Explain.
4. For a 5-12-13 triangle, calculate sine and cosine of one acute angle.
5. Use scale factor 2.5 and verify one trig ratio remains constant.
6. Explain how constant ratios make indirect measurement possible.

A student says sine is ‘opposite divided by hypotenuse’ only because calculators are programmed that way. Construct a geometric argument using similar triangles showing the ratio is a property of the angle itself.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

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.

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.

• I can justify trig ratios using similarity.
• I can label sides relative to a reference angle.
• I can show scale factors cancel in ratios.
• I can connect SOH-CAH-TOA to geometry.
• I can explain indirect measurement conceptually.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

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.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9M03 — Scale, Similarity, Pythagoras and Right-Triangle Trigonometry
• AC9M9SP02 — Enlargement, Similarity, Ratio and Scale

• Australian Curriculum v9.0 — Mathematics
• Victorian Curriculum F–10 Version 2.0 — Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for The constancy of the sine, cosine and tangent ratios for...

Mapped skill: recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.