ACARA v9 CONTENT DESCRIPTION “recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity”
Builds on: Pythagoras and Trigonometry in Right-Angled Triangles (AC9M9M03). This unit builds on similarity and scale, and on the trigonometric ratios introduced in the measurement strand. Understanding why those ratios are constant is the foundation for all later trigonometry and for the enlargement transformation studied next in this strand.
A quiet miracle of fixed ratios
Trigonometry rests on a quiet miracle: for a fixed angle in a right-angled triangle, the ratios of the sides never change, no matter how large or small the triangle is. The sine of thirty degrees is one half whether the triangle fits on a fingernail or spans a football field. This unit explains why that constancy holds, and the explanation is entirely about similar triangles. Once you see it, the sine, cosine and tangent stop looking like arbitrary button presses on a calculator and become natural consequences of shape.
Same angle, three sizes
Three right-angled triangles share one acute angle but differ in size; because the angle is fixed they are the same shape, and every side ratio is the same in all three.
The small triangle has opp / adj = 0.577. Pick any size: the angle is 30° in all three, so this ratio is the same number every time. Fixing the angle fixes the shape, and a fixed shape forces every side ratio to be constant no matter how large or small the triangle is drawn.
What makes triangles similar
Begin with what similar triangles are. Two triangles are similar when they have the same angles; they are the same shape but possibly different sizes. A crucial property follows: corresponding sides of similar triangles are in proportion. If one triangle is an enlargement of another by a scale factor of two, then every side of the larger is exactly twice the matching side of the smaller. The shape is preserved; only the size changes.
Similar means proportional sides
Scaling a 3-4-5 triangle keeps every angle identical and multiplies each side by the same scale factor; choose the factor and watch the sides scale together.
Both triangles have identical angles, so they are similar. With scale factor 2, each side of the copy is its match in the 3-4-5 triangle times 2: 3 x 2 = 6, 4 x 2 = 8, 5 x 2 = 10. The shape is preserved while the size changes.
Enlarging a triangle keeps the ratios
Now fix an acute angle, say the angle in a three, four, five right-angled triangle. Double every side and you get a six, eight, ten triangle, which has exactly the same angles and so is similar. Compare the ratio of the side opposite the angle to the hypotenuse. In the small triangle it is three over five; in the large one it is six over ten. These are equal, both nought point six. The same happens for every ratio: four over five equals eight over ten, and three over four equals six over eight. Enlarging the triangle changed every length but left every ratio untouched.
The ratio is unchanged
Whichever side ratio you choose, the small triangle and the doubled triangle give the same value; enlarging changes the lengths but never the ratio.
opp / hyp is 3/5 in the small triangle and 6/10 in the large one, and both equal 0.6. Try each ratio: every one agrees across the two sizes. Enlarging changed every length but left the ratios untouched, which is exactly what constancy means.
Why the scale factor cancels
The reason is simple algebra. Suppose the larger triangle is the smaller one scaled by a factor k, so each side is k times the corresponding small side. The ratio of opposite to hypotenuse in the large triangle is k times the opposite, over k times the hypotenuse. The factor k appears on top and bottom and cancels, leaving exactly the ratio from the small triangle. Because any two right-angled triangles sharing the same acute angle are similar, this cancellation always happens, and the ratio depends only on the angle.
Why k cancels
When a triangle is enlarged by a factor k, both the opposite and the hypotenuse are multiplied by k; the k appears top and bottom and cancels, leaving the original ratio.
Enlarging multiplies every side by k, so opposite over hypotenuse becomes (k x opp) / (k x hyp). The k appears in both numerator and denominator and cancels, leaving opp / hyp. That is why the ratio depends only on the angle, not on the size.
Naming the constant ratios
This constant ratio is precisely what we name. For a given acute angle, the ratio of the opposite side to the hypotenuse is the sine of that angle, the ratio of the adjacent side to the hypotenuse is the cosine, and the ratio of the opposite to the adjacent is the tangent. Because of the similarity argument, each of these is a single fixed number for each angle, which is exactly why a calculator can store them. The angle of thirty-six point eight seven degrees in any three, four, five style triangle always has a tangent of nought point seven five.
Naming sine, cosine, tangent
Each constant ratio gets a name: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.
For the marked angle, sine = opposite / hypotenuse, which is 3/5 = 0.6 here. The two highlighted sides are exactly the ones in the ratio. Because of the similarity argument each name is a single fixed number for that angle, which is exactly why a calculator can store it.
Exact and irrational values
Some of these values are clean and exact, while others are irrational. The sine of thirty degrees is exactly one half, and the tangent of forty-five degrees is exactly one, because a forty-five degree right-angled triangle has two equal sides. Others run forever without repeating: the cosine of thirty degrees is the square root of three over two, approximately nought point eight six six but never exactly that decimal, and the sine of forty-five degrees is approximately nought point seven zero seven. Treating these irrational values as exact surds, rather than rounded decimals, keeps later calculations honest.
Exact vs irrational ratios
Some special ratios are exact equalities, while others are irrational and only approximated by a decimal; the two cases are written differently on purpose.
sin 30° = 1/2 and tan 45° = 1 are exact equalities. cos 30° = √3 over 2 ≈ 0.866 and sin 45° ≈ 0.707 are irrational: the decimals only approximate them, so they take ~ and never an equals sign.
Why constancy makes trigonometry work
The payoff of constancy is enormous. Because each angle has fixed ratios, a single table or calculator can solve any right-angled triangle, of any size, once the angle is known. Surveyors, navigators and builders rely on this every day: measure one angle and one length, and the constant ratios deliver every other length. The deep idea, that similarity forces the side ratios to depend on the angle alone, is what makes trigonometry both possible and trustworthy, turning the geometry of similar shapes into a precise tool for measurement.
Quick self-check
1. A 3-4-5 right triangle is enlarged to a 6-8-10 triangle. What happens to the tangent of the marked angle?
2. Two right triangles share the same acute angle but one is three times larger. How do their sine ratios for that angle compare?
3. Why does the scale factor k cancel in the ratio opposite/hypotenuse?
4. Which of these is an EXACT value (not an approximation)?
5. In a 5-12-13 right triangle, what is the sine of the angle opposite the side of length 5?