Choose the method from what you are given. Two sides and a missing third side means Pythagoras. One side and an angle, or two sides and a missing angle, means trigonometry.
Some questions need both in sequence, and then the skill is keeping your working accurate between the steps.
This lesson builds on choosing the ratio and finding missing angles within right-triangle trigonometry.
How do you pick the method?
| You are given | You want | Method |
|---|---|---|
| Two sides | third side | Pythagoras: a² + b² = c² |
| One side and one angle | another side | sin, cos or tan |
| Two sides | an angle | inverse sin, cos or tan |
| One side and one angle | two other sides | trigonometry for one, then either method for the other |
A longer question can hide a chain of these rows. Write one line saying what you can find next, then do it, then look again at what you now know.
How do you handle a two-step question?
- Sketch and label what you know.
- List what you can find straight away.
- Find it with the shortest method and keep the full calculator value.
- Use that value in the next step.
- Round only at the end, to the accuracy asked.
- Check with the other relationship if the question allows.
Worked example
Triangle ABC has a right angle at B. AB = 20 cm and angle A = 32°. Find BC and then AC to 3 significant figures.
Step 1, label from A: AB is adjacent (20 cm), BC is opposite, and AC is the hypotenuse.
Step 2, find BC with tangent: opposite and adjacent, so tan 32° = BC/20, which gives BC = 20 × tan 32° = 12.497… cm.
Step 3, find AC with Pythagoras: AC² = 20² + 12.497² = 400 + 156.18 = 556.18, so AC = 23.58… cm.
Step 4, round: BC = 12.5 cm and AC = 23.6 cm (3 s.f.).
Check: AC is the longest side, as it should be. Using cosine, AC = 20 ÷ cos 32° = 20 ÷ 0.8480 = 23.58 cm, which agrees.
A one-step case, for contrast: if the legs are 8 cm and 15 cm, Pythagoras gives 64 + 225 = 289, so the hypotenuse is 17 cm. The angle opposite the 8 cm leg is sin⁻¹(8/17) = 28.1°. Trigonometry only entered when an angle was asked for.
The mistake to watch for
A common slip is rounding the first answer and carrying it forward.
Mistaken working: the student rounds BC to 12 cm, then AC² = 400 + 144 = 544, so AC = 23.3 cm.
The correct value is 23.6 cm. Rounding BC from 12.497 to 12 was a large change.
The correction is to keep 12.497… on the calculator (use the answer key or memory) and to round only the final value. Even rounding to 12.5 is safer than 12, but the safest habit is not to round at all until the end.
Check yourself
1. A right-angled triangle has one leg of 7 cm and a hypotenuse of 25 cm. Find the other leg, and the angle opposite the 7 cm leg to 1 decimal place.
Show answer
Other leg: √(25² − 7²) = √(625 − 49) = √576 = 24 cm. Angle: sin θ = 7/25, so θ = sin⁻¹(0.28) = 16.3°.
2. A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse and the angle opposite the 6 cm leg.
Show answer
Hypotenuse: √(36 + 64) = √100 = 10 cm. Angle: tan θ = 6/8 = 0.75, so θ = 36.9°.
3. The hypotenuse is 13 cm and one angle is 62°. Find both other sides to 3 significant figures, then check them.
Show answer
Opposite 62°: 13 × sin 62° = 11.5 cm. Adjacent: 13 × cos 62° = 6.10 cm. Check: 11.478² + 6.103² = 131.75 + 37.25 = 169 = 13². ✓
Where this leads next
Next is interpreting an angle of elevation diagram, where a sentence has to be turned into a triangle first. Practise the mix in the right-triangle trigonometry practice set. The triangle and bearings reasoning board asks you to choose the relationship before any calculation, and the non-calculator working trainer covers squares and square roots.
Students who see the method but rush the choice lose marks at the very first line. Our teachers work on that in online one-to-one Mathematics tuition.