Mathematics 7–10 · Year 9

Why the trigonometric ratios are constant: measuring similar right-angled triangles

Measurement and space

Practical, model not builtLow risk

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The idea

Right-angled triangles that share one acute angle are similar, so the ratio opposite over adjacent is the same for all of them, and that shared number is the tangent of the angle.

What you need

A3 paper, ruler, set square, protractor, sharp pencil
Calculator with trigonometric functions

How to do it

Draw a 35 degree angle with its vertex at the left edge of the page and the lower arm horizontal, 25 cm long.
At 5, 10, 15 and 20 cm along the lower arm draw perpendiculars up to the upper arm, making four right-angled triangles that share the 35 degree angle.
Measure the opposite side (the perpendicular) and the hypotenuse of each triangle to the nearest millimetre.
Compute opposite/adjacent, opposite/hypotenuse and adjacent/hypotenuse for all four triangles and tabulate.
Compare each column with the calculator's tan 35, sin 35 and cos 35, then repeat the drawing for 50 degrees.

What you should see

For 35 degrees the four values of opposite/adjacent all lie near 0.700 (tan 35° = 0.7002), opposite/hypotenuse near 0.574 (sin 35° = 0.5736) and adjacent/hypotenuse near 0.819 (cos 35° = 0.8192). A 1 mm error in the measured opposite side changes opposite/adjacent by 0.1 ÷ 5 = 0.02 in the 5 cm triangle and by 0.1 ÷ 20 = 0.005 in the 20 cm triangle, so the larger triangles give the closer agreement. The learner knows it worked when the four ratios in each column agree with each other and with the calculator to two decimal places for the larger triangles.

What changes

What you change: size of the triangle (adjacent side)
What you measure: ratio of sides
What you keep the same: same acute angle
perpendiculars drawn with a set square
same ruler

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A bigger triangle has a bigger sine.
Sine is a length rather than a ratio.
The ratios only apply to triangles drawn in one orientation.

Safety card

Low riskLearners carry it out

Hazards

No hazard is listed.

Controls

No control is listed.

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K–10 Syllabus (2022), Stage 5 Trigonometry A; page read 2026-09-22MA5-TRG-C-01
Mathematics K–10 Syllabus (2022), Stage 5 Properties of geometrical figures A; page read 2026-09-22MA5-GEO-C-01
Australian Curriculum v9AC9M9SP01AC9M9M03

Sources

The pages the author read to write this activity.

All Concept Studio activities