Choosing the right ratio means naming three sides before you calculate: the hypotenuse, the side opposite your angle and the side adjacent to it. Then look at which two sides the question involves, one known and one wanted, and match them to sine, cosine or tangent.
This is the first skill in right-triangle trigonometry, and every later lesson in the module relies on it.
Why does the angle you start from matter?
The hypotenuse is always the longest side, opposite the right angle. It never changes. Opposite and adjacent depend on the angle you pick: the side opposite one acute angle is the side adjacent to the other.
So the first step is always to put a finger on the angle you are using, then label the sides from that angle.
How do you choose the ratio, step by step?
- Draw or redraw the triangle and mark the right angle.
- Mark the angle you know or want.
- Label the three sides: hypotenuse (H), opposite (O) and adjacent (A).
- Underline the two sides in the question: the one you have and the one you want.
- Match the pair:
| Sides involved | Ratio | Formula |
|---|---|---|
| Opposite and hypotenuse | sine | sin θ = O/H |
| Adjacent and hypotenuse | cosine | cos θ = A/H |
| Opposite and adjacent | tangent | tan θ = O/A |
- Write the equation, then rearrange to find the unknown.
Worked example
In a right-angled triangle, the angle at A is 35° and the hypotenuse is 14 cm. Find the side opposite angle A, which we call x.
Step 1, label from 35°: the hypotenuse is 14 cm. The side we want, x, is opposite the angle. The adjacent side is not mentioned, so ignore it.
Step 2, match the pair: opposite and hypotenuse means sine.
Step 3, write the equation: sin 35° = x/14.
Step 4, rearrange: x = 14 × sin 35°.
Step 5, calculate: sin 35° = 0.5736, so x = 14 × 0.5736 = 8.03 cm (3 s.f.).
Check: x is shorter than the hypotenuse, 14 cm, as every leg must be. Good.
A second case shows tangent. If the angle is 40° and the adjacent side is 9 cm, the opposite side is 9 × tan 40° = 9 × 0.8391 = 7.55 cm (3 s.f.), because opposite and adjacent point to tangent.
The mistake to watch for
A common slip is to label the sides from the wrong angle, usually the other acute angle, and then pick the wrong ratio.
Mistaken working: the student treats the side next to the 35° angle as “opposite”, writes cos 35° = x/14 and gets 14 × 0.8192 = 11.5 cm.
The side x was opposite 35°, so the equation should use sine. The answer 11.5 cm is not the side we wanted.
The correction is to put a finger on the angle first, then trace to the side directly across from it. The side that touches the angle, apart from the hypotenuse, is the adjacent one.
Check yourself
Try these, then open each answer.
1. The angle is 60° and the hypotenuse is 20 cm. Find the adjacent side.
Show answer
Adjacent and hypotenuse means cosine. x = 20 × cos 60° = 20 × 0.5 = 10 cm.
2. The angle is 28° and the hypotenuse is 15 cm. Find the opposite side to 3 significant figures.
Show answer
Opposite and hypotenuse means sine. x = 15 × sin 28° = 15 × 0.4695 = 7.04 cm.
3. The angle is 42° and the adjacent side is 7.5 cm. Find the hypotenuse to 3 significant figures.
Show answer
Adjacent and hypotenuse means cosine: cos 42° = 7.5/H, so H = 7.5 ÷ cos 42° = 7.5 ÷ 0.7431 = 10.1 cm.
Where this leads next
Once naming the sides becomes automatic, move on to finding a missing angle with the correct mode, then test the whole topic with the right-triangle trigonometry practice set. The non-calculator working trainer is useful for checking the fraction arithmetic in ratios such as 5/13.
Some students follow each example but still choose the wrong ratio when the diagram is turned around. That is the kind of pattern our teachers look for in online one-to-one Mathematics tuition.