Labelling Sides & the Trig Ratios
Learn to label the sides of a right-angled triangle and use the trigonometric ratios for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-04). Before using trigonometry, you name each side from the marked angle, which tells you which ratio to use.
You will pick out the hypotenuse, opposite and adjacent sides, name sides by their letters, and use SOH CAH TOA to choose sine, cosine or tangent for the two sides you are working with — the first step in finding unknown sides and angles in ramp, roof and ladder problems.
Theory
Before using trigonometry, name the three sides of a right-angled triangle from the marked angle: the hypotenuse, opposite and adjacent. This Year 12 Mathematics Standard 1 (NSW) guide shows how to label the sides and use SOH CAH TOA to choose sine, cosine or tangent for the two sides you are working with.
In a right-angled triangle the hypotenuse (hyp) is opposite the right angle; it is the longest side and it never changes. The other two sides are named from the marked angle \(\theta\).
The opposite side (opp) is the side that does not touch \(\theta\). The adjacent side (adj) touches \(\theta\) and is not the hypotenuse. Sides can also be named by letters: the side opposite an angle joins the other two vertices, e.g. in \(\triangle PRT\) the side opposite \(R\) is \(PT\).
The three trig ratios link pairs of sides: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}}\) (SOH), \(\cos\theta=\dfrac{\text{adj}}{\text{hyp}}\) (CAH) and \(\tan\theta=\dfrac{\text{opp}}{\text{adj}}\) (TOA). Labelling the sides sets up ramp, roof and ladder problems so you can later find a missing side or angle.
The three trigonometric ratios (remember SOH CAH TOA):
Pick the ratio that uses the two sides in the question:
| Sides you have | Ratio | Memory aid |
|---|---|---|
| opp and hyp | sine | SOH |
| adj and hyp | cosine | CAH |
| opp and adj | tangent | TOA |
How to label a triangle and choose the ratio
- Find the hypotenuse: the side opposite the right-angle box.
- Label from the marked angle: the side not touching \(\theta\) is opposite; the other side touching \(\theta\) is adjacent.
- Choose the ratio that uses the two sides in the question (SOH CAH TOA) and write it the right way up.
Sketch the triangle. Start from the right angle at \(M\), then look from \(L\).
(i) Opposite the right angle at \(M\) \(\Rightarrow\) \(LN\).
(ii) The side not touching \(L\) \(\Rightarrow\) \(MN\).
(iii) Touches \(L\), not the hypotenuse \(\Rightarrow\) \(LM\).
Hyp \(LN\), opp \(MN\), adj \(LM\).
Sketch the triangle and label from \(\theta\): opp \(=8\), adj \(=15\), hyp \(=17\).
| \(\sin\theta\) | \(=\) | \(\dfrac{8}{17}=0.470\ldots\approx0.47\) |
| \(\cos\theta\) | \(=\) | \(\dfrac{15}{17}=0.882\ldots\approx0.88\) |
| \(\tan\theta\) | \(=\) | \(\dfrac{8}{15}=0.533\ldots\approx0.53\) |
\(0.47\), \(0.88\), \(0.53\).
Sketch the triangle. From \(25^\circ\): \(x\) is the opposite side and \(14\) cm is the adjacent side. Opposite and adjacent \(\to\) TOA.
| \(\tan\theta\) | \(=\) | \(\dfrac{\text{opp}}{\text{adj}}\) |
| \(\tan25^\circ\) | \(=\) | \(\dfrac{x}{14}\) |
Use \(\tan25^\circ=\dfrac{x}{14}\).
Match the pair of sides to SOH CAH TOA.
| \(\text{(i) SOH: }\sin\theta\) | \(=\) | \(\dfrac{7}{25}=0.28\) |
| \(\text{(ii) CAH: }\cos\theta\) | \(=\) | \(\dfrac{24}{25}=0.96\) |
| \(\text{(iii) TOA: }\tan\theta\) | \(=\) | \(\dfrac{7}{24}=0.291\ldots\approx0.29\) |
(i) sine \(0.28\); (ii) cosine \(0.96\); (iii) tangent \(0.29\).
Common pitfalls
Frequently asked questions
How do you label the sides of a right-angled triangle?
First find the hypotenuse, the side opposite the right angle. Then, from the marked angle, the side that does not touch the angle is the opposite side and the other side that touches it is the adjacent side.
What does SOH CAH TOA mean?
It is a memory aid for the three trig ratios: sine equals opposite over hypotenuse (SOH), cosine equals adjacent over hypotenuse (CAH), and tangent equals opposite over adjacent (TOA).
How do I know whether to use sin, cos or tan?
Look at the two sides in the question, labelled from the angle. Opposite and hypotenuse means sine, adjacent and hypotenuse means cosine, and opposite and adjacent means tangent.
What is the difference between the opposite and adjacent side?
Both are named from the marked angle. The opposite side does not touch the angle; the adjacent side touches the angle and is not the hypotenuse. If the marked angle moves to the other acute angle, they swap.
Can sin or cos be greater than 1?
No. In a right-angled triangle the hypotenuse is the longest side, so opposite over hypotenuse and adjacent over hypotenuse are always less than 1. Tangent can be greater than 1.