The Sine and Cosine Rules

Beyond right-angled triangles

Basic trigonometry with SOH-CAH-TOA only works for right-angled triangles. But most real triangles have no right angle. The sine rule and cosine rule extend trigonometry to any triangle, letting you find missing sides and angles whatever the shape. If you have not met sin, cos and tan yet, start with introduction to trigonometry.

Labelling a triangle

There is one essential convention. Label each angle with a capital letter (A, B, C) and label each side with the lowercase letter of the angle it is opposite to:

• Side a is opposite angle A.
• Side b is opposite angle B.
• Side c is opposite angle C.

Getting this matching right is the whole battle — both rules depend on pairing each side with the angle facing it.

The sine rule

The sine rule (or law of sines) states:

a / sin A = b / sin B = c / sin C

You can also flip it (handy when finding angles):

sin A / a = sin B / b = sin C / c

Use the sine rule when you have a complete side–angle pair (a side and the angle opposite it), plus one more piece of information. Typical cases: two angles and any side (AAS/ASA), or two sides and an angle opposite one of them.

Finding a missing side

Worked example 1: In triangle ABC, angle A = 40°, angle B = 75°, and side a = 8 cm. Find side b.

1. We have the pair a and A, and we want b with its angle B — a matching pair, so use the sine rule.
2. Set up: b / sin B = a / sin A, so b / sin 75° = 8 / sin 40°.
3. Multiply both sides by sin 75°: b = 8 × sin 75° / sin 40°.
4. Evaluate: sin 75° ≈ 0.9659, sin 40° ≈ 0.6428.
5. b = 8 × 0.9659 / 0.6428 ≈ 7.727 / 0.6428 ≈ 12.0 cm.

Finding a missing angle

Worked example 2: In triangle ABC, side a = 10 cm, side b = 7 cm, and angle A = 50°. Find angle B.

1. We know the pair a and A, and side b, so use the flipped sine rule.
2. sin B / b = sin A / a, so sin B / 7 = sin 50° / 10.
3. sin B = 7 × sin 50° / 10 = 7 × 0.766 / 10 = 0.536.
4. B = sin⁻¹(0.536) ≈ 32.4°.

The ambiguous case: Because sin 32.4° = sin 147.6°, an angle could in principle be obtuse. Here, since side b is shorter than side a, angle B must be smaller than angle A, so the acute answer 32.4° is correct. Always sanity-check against the triangle.

The cosine rule

When you do not have a matching side–angle pair, the sine rule cannot start. The cosine rule (law of cosines) handles these cases:

a² = b² + c² − 2bc · cos A

There are matching versions for the other angles, e.g. b² = a² + c² − 2ac · cos B.

Use the cosine rule when you have:

• SAS — two sides and the included angle (the angle between them), to find the third side, or
• SSS — all three sides, to find any angle (using the rearranged form below).

Finding a side (SAS)

Worked example 3: In triangle ABC, b = 6 cm, c = 9 cm, and the included angle A = 60°. Find side a.

1. Apply: a² = b² + c² − 2bc · cos A.
2. Substitute: a² = 6² + 9² − 2(6)(9) · cos 60°.
3. cos 60° = 0.5, so a² = 36 + 81 − 108 × 0.5 = 117 − 54 = 63.
4. a = √63 ≈ 7.94 cm.

Finding an angle (SSS)

Rearrange the cosine rule to isolate the angle:

cos A = (b² + c² − a²) / (2bc)

Worked example 4: A triangle has sides a = 8, b = 5, c = 7. Find angle A.

1. cos A = (b² + c² − a²) / (2bc) = (25 + 49 − 64) / (2 × 5 × 7).
2. = (74 − 64) / 70 = 10 / 70 ≈ 0.1429.
3. A = cos⁻¹(0.1429) ≈ 81.8°.

The connection to Pythagoras

The cosine rule is a generalisation of the Pythagorean theorem. When angle A = 90°, cos 90° = 0, so the term −2bc·cos A disappears and the rule becomes:

a² = b² + c²

— exactly Pythagoras. So Pythagoras is just the special right-angled case of the cosine rule.

Area of any triangle

When you know two sides and the angle between them, the area is:

Area = ½ × a × b × sin C

(any two sides and the angle between them).

Worked example 5: Find the area of a triangle with sides a = 12 cm, b = 9 cm, and included angle C = 50°.

1. Area = ½ × 12 × 9 × sin 50°.
2. sin 50° ≈ 0.766.
3. Area = 54 × 0.766 ≈ 41.4 cm².

A multi-step worked problem

Real questions often need two rules in sequence. Find each missing part in order.

Worked example 6: In triangle ABC, b = 10 cm, c = 14 cm, and angle A = 55°. Find side a, then angle B, then angle C.

1. Side a (cosine rule, SAS): a² = b² + c² − 2bc·cos A = 10² + 14² − 2(10)(14)cos 55°.
2. cos 55° ≈ 0.5736, so a² = 100 + 196 − 280(0.5736) = 296 − 160.6 = 135.4, giving a ≈ 11.6 cm.
3. Angle B (sine rule): now we have the pair a and A. sin B / b = sin A / a, so sin B = 10 × sin 55° / 11.6 = 10 × 0.8192 / 11.6 ≈ 0.706.
4. B = sin⁻¹(0.706) ≈ 44.9°.
5. Angle C: angles in a triangle sum to 180°, so C = 180° − 55° − 44.9° = 80.1°.

Notice the strategy: use the cosine rule to crack open the triangle (it needs no matching pair), then switch to the faster sine rule, and finish with the angle sum.

Choosing the right rule

You know…	Want	Use
2 angles + 1 side	A side	Sine rule
2 sides + 1 opposite angle	An angle	Sine rule
2 sides + included angle (SAS)	Third side	Cosine rule
3 sides (SSS)	An angle	Cosine rule (rearranged)
2 sides + included angle	Area	½ ab·sin C

Where these rules are used

Surveyors and navigators use the sine and cosine rules to find distances they cannot measure directly — across a river, between ships, or between landmarks — a technique called triangulation. Astronomers use them to compute distances between stars. Engineers use them to resolve forces in frameworks that are not right-angled. Any time three points form a triangle and you can measure some sides or angles, these rules unlock the rest.

Practice activity

Use the correct rule for each.

1. Triangle with A = 35°, B = 65°, side a = 9 cm. Find side b.
2. Triangle with sides b = 5, c = 8, included angle A = 70°. Find side a.
3. Triangle with sides 6, 7, 9. Find the largest angle.
4. Find the area of a triangle with sides 10 and 14 and included angle 30°.

Answers:

1. Sine rule: b = 9 × sin 65° / sin 35° = 9 × 0.9063 / 0.5736 ≈ 14.2 cm.
2. Cosine rule: a² = 25 + 64 − 2(5)(8)cos 70° = 89 − 80(0.342) = 89 − 27.4 = 61.6, so a ≈ 7.85.
3. Largest angle faces the longest side (9). cos θ = (36 + 49 − 81)/(2·6·7) = 4/84 ≈ 0.0476, θ ≈ 87.3°.
4. Area = ½ × 10 × 14 × sin 30° = 70 × 0.5 = 35 square units.

Summary

The sine rule a/sin A = b/sin B = c/sin C and the cosine rule a² = b² + c² − 2bc·cos A extend trigonometry to any triangle. Label sides opposite their angles, then choose: the sine rule when you have a side–angle pair, and the cosine rule for SAS (find a side) or SSS (find an angle). The cosine rule reduces to Pythagoras at 90°, and the area of any triangle is ½ ab·sin C. Watch for the ambiguous case when finding angles with the sine rule.