What is the law of sines?
Label a triangle ABC. The side facing angle A is called a. The side facing B is b. The side facing C is c.
The law of sines (also called the sine rule) says:
a / sin A = b / sin B = c / sin C
You can also flip every fraction: sin A / a = sin B / b = sin C / c. Flipping is handy when you want to find an angle.
It works in every triangle: acute, right or obtuse. In a right triangle with C = 90°, sin C = 1, so c is just the hypotenuse and the rule turns into sin A = a / c, the ratio you already know.
Why it is true: the proof with a height
Drop a perpendicular (a height) h from C to side AB.
- In the left right-angled triangle, sin A = h / b, so h = b sin A.
- In the right right-angled triangle, sin B = h / a, so h = a sin B.
Both describe the same height, so b sin A = a sin B. Divide both sides by sin A sin B to get a / sin A = b / sin B. Dropping a height from A instead gives b / sin B = c / sin C the same way.
The common ratio is 2R
Draw the circle that passes through A, B and C (the circumcircle, radius R). One can show that each ratio equals the diameter: a / sin A = 2R. So the sine rule also lets you find the circumradius.
Area link
Area = ½ × base × height = ½ × c × (b sin A) = ½ bc sin A. This formula is often taught next to the sine rule.
Solving triangles with the sine rule (AAS and ASA)
To solve a triangle means to find all three sides and all three angles.
- If you know two angles, find the third: A + B + C = 180°.
- Pick a pair you know completely (a side and its opposite angle).
- Set up one fraction for that pair and one for the side you want. Cross-multiply.
Example: A = 40°, B = 60°, c = 10 cm. Then C = 80°, and a = 10 × sin 40° ÷ sin 80° ≈ 6.53 cm, b = 10 × sin 60° ÷ sin 80° ≈ 8.79 cm.
Check: the biggest angle (80°) faces the biggest side (10 cm). Always do this quick check.
The ambiguous case (SSA)
If you know two sides and an angle that is not between them, the answer may not be unique. Say you know A, a and b. Work out sin B = b sin A / a.
- sin B > 1: no triangle. Side a is too short to reach the base.
- sin B = 1: one right triangle.
- sin B < 1: B could be the acute angle B₁ or the obtuse angle 180° − B₁. Keep the obtuse one only if A + (180° − B₁) < 180°. If a ≥ b, only the acute one works.
In the 3D, side a swings like a door and touches the base at two points: that is why there can be two triangles.
Exam tips and where the sine rule is used
- Use the sine rule when you have a full pair (side + opposite angle). Use the cosine rule when you have SAS or SSS.
- Keep your calculator in degree mode.
- Do not round too early: keep 3–4 decimals until the last line.
- Uses: surveying, navigation, finding heights of towers and hills, astronomy, GPS and phone-tower location.
Key formulas and definitions
- a / sin A = b / sin B = c / sin C = 2R
- sin A / a = sin B / b = sin C / c (use to find an angle)
- A + B + C = 180°
- Area = ½ bc sin A = ½ ca sin B = ½ ab sin C
- SSA test: sin B = b sin A / a (>1 none, =1 one right triangle, <1 maybe two)
Worked examples
1. In triangle ABC, A = 30°, B = 45° and a = 8 cm. Find b.
b / sin 45° = 8 / sin 30°. So b = 8 × sin 45° ÷ sin 30° = 8 × 0.7071 ÷ 0.5 ≈ 11.31 cm.
2. A = 40°, B = 60°, c = 10 cm. Solve the triangle.
C = 180° − 40° − 60° = 80°. a = 10 sin 40° ÷ sin 80° = 6.428 ÷ 0.9848 ≈ 6.53 cm. b = 10 sin 60° ÷ sin 80° = 8.660 ÷ 0.9848 ≈ 8.79 cm.
3. a = 12, b = 9, A = 70°. Find B.
sin B = 9 sin 70° ÷ 12 = 9 × 0.9397 ÷ 12 ≈ 0.7048. B ≈ 44.8°. The obtuse choice 135.2° is too big (70° + 135.2° > 180°), so only one triangle.
4. A = 30°, a = 6, b = 10. How many triangles? Find B.
sin B = 10 × 0.5 ÷ 6 ≈ 0.8333. B₁ ≈ 56.4° or B₂ ≈ 123.6°. Both fit with A = 30° (sums 86.4° and 153.6° are below 180°), so two triangles.
5. A = 50°, a = 5, b = 10. How many triangles?
sin B = 10 sin 50° ÷ 5 = 2 × 0.766 = 1.532. A sine cannot be more than 1, so no triangle exists.
6. Two points P and Q are 100 m apart on a riverbank. A tree T is on the far bank. Angle TPQ = 60° and angle TQP = 70°. Find PT.
Angle T = 180° − 60° − 70° = 50°. PT faces angle Q, PQ faces angle T. PT = 100 × sin 70° ÷ sin 50° = 93.97 ÷ 0.766 ≈ 122.7 m.
7. In a triangle, a = 7 cm and A = 30°. Find the radius R of the circumcircle.
a / sin A = 2R. 7 / 0.5 = 14 = 2R, so R = 7 cm.
Common mistakes
- Pairing a side with the wrong angle. Side a must go with angle A, the angle facing it, not an angle touching it.
- Forgetting the second answer in SSA: sin B = 0.8 gives B ≈ 53.1° or B ≈ 126.9°. Test both.
- Calculator in radian mode: sin 30 would give −0.988 instead of 0.5.
- Using the sine rule for SAS or SSS. With no complete side–angle pair, start with the cosine rule.