Year 9AC9M9SP01

Why the Trig Ratios Are Constant

Recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity.

Every right triangle with the same angle is similar to every other. That is the whole reason a calculator can store one number for sin 30.

Method

Similar triangles have equal ratios

Same angles means matching sides are in the same ratio.

The ratio depends only on the angle

\sin\theta = \frac{O}{H}

Not on how big the triangle is.

That is why the tables work

One value of sin 30 serves every triangle in the world.

Label from the angle you are using

Opposite and adjacent swap if you move to the other angle.

e.g.

\text{A triangle has sides } 3, 4, 5. \text{ Another has } 6, 8, 10. \text{ Show the ratios match.}

Check the triangles are similar. $6:3 = 8:4 = 10:5 = 2:1 \quad\checkmark$
Take the angle opposite the shortest side in the first. $\sin\theta = \frac{3}{5} = 0{,}6$
Take the matching angle in the second. $\sin\theta = \frac{6}{10} = 0{,}6$
The ratios are identical, though the triangles differ in size. $0{,}6 = 0{,}6$
Conclude the reason. $\text{the ratio depends on the angle alone}$

Check similarity first, then compare the ratios.

\text{Triangle } 5,12,13. \text{ Find } \tan\theta \text{ for the angle opposite the 5.}

Answer

\frac{5}{12}

\text{Triangle } 10,24,26. \text{ Find the same ratio.}

Answer

\frac{10}{24} = \frac{5}{12}

\text{Why do both give the same answer?}

Answer

\text{The triangles are similar.}

\text{Does doubling every side change } \cos\theta?

Answer

\text{No.}

\text{Find } \sin 30^\circ

Answer

0{,}5

\text{Why can a calculator store one value per angle?}

Answer

\text{The ratio never changes for that angle.}