Use the sine rule when you know a side and the angle opposite it. Use the cosine rule when you know two sides and the angle between them, or all three sides.
This lesson is part of solution of triangles. Read the given information first, and only then write a formula.
What does the question give you?
Label the triangle with sides a, b, c opposite angles A, B, C. Then sort the given data into one of four patterns.
| Given | Name | Rule to start with |
|---|---|---|
| Two angles and any side | AAS or ASA | Sine rule |
| Two sides and the angle between them | SAS | Cosine rule |
| All three sides | SSS | Cosine rule |
| Two sides and an angle not between them | SSA | Sine rule (check the ambiguous case) |
The test for the sine rule is simple: you must already know one complete pair, a side with its opposite angle.
Worked example 1: two sides and the angle between them
In triangle ABC, AB = 7 cm, AC = 9 cm and angle A = 60°. Find BC.
The known angle A is between AB and AC, and BC is opposite A. There is no known side-angle pair, so use the cosine rule:
BC² = 7² + 9² − 2(7)(9) cos 60° = 49 + 81 − 126(0.5) = 130 − 63 = 67
BC = √67 ≈ 8.19 cm.
Worked example 2: two angles and a side
In triangle PQR, angle P = 48°, angle Q = 65° and QR = 12 cm. Find PR.
QR is opposite P, so the pair (48°, 12 cm) is known. PR is opposite Q. Use the sine rule:
PR ÷ sin 65° = 12 ÷ sin 48°
PR = 12 sin 65° ÷ sin 48° = 10.876 ÷ 0.7431 ≈ 14.6 cm.
Worked example 3: three sides
A triangle has sides 5 cm, 6 cm and 7 cm. Find the angle opposite the 7 cm side.
Let the 7 cm side be c. Then cos C = (5² + 6² − 7²) ÷ (2 × 5 × 6) = 12 ÷ 60 = 0.2, so C ≈ 78.5°.
The mistake that costs marks
The common slip is to reach for the sine rule on a SAS question because it looks shorter. For example 1, a student writes BC ÷ sin 60° = 7 ÷ sin C, but sin C is unknown and so is BC. The equation has two unknowns and cannot be solved.
| Step | Wrong | Right |
|---|---|---|
| Check for a pair | (skipped) | Is any side known with its opposite angle? |
| Answer | No pair, yet sine rule used | No pair, so cosine rule |
| Result | Stuck with two unknowns | BC ≈ 8.19 cm |
Write one line naming the pattern (SAS, AAS and so on) before any substitution. That habit helps you avoid wrong starts.
Check yourself
In triangle KLM, KL = 10 cm, angle K = 35° and angle L = 80°. Find LM. Which rule do you use, and what must you find first?
Answer
Two angles and a side are given, so use the sine rule. LM is opposite K, but KL is opposite M, so first find M: 180° − 35° − 80° = 65°.
LM ÷ sin 35° = 10 ÷ sin 65°
LM = 10 sin 35° ÷ sin 65° = 5.736 ÷ 0.9063 ≈ 6.33 cm.
What to study next
Some sine-rule questions give an angle that is not between the known sides and can give two answers. Continue with the ambiguous sine-rule case, then log your slips with the mistake log.
If you want a teacher to work through triangle questions with you, see online one-to-one Additional Mathematics tuition.