The sine rule (law of sines)

The sine rule, called the law of sines in the US, says that in any triangle each side divided by the sine of its opposite angle gives the same answer: a/sin A = b/sin B = c/sin C. Use it to find a missing side or angle in a triangle without a right angle, whenever you know a side and the angle opposite it, plus one more side or angle.

The sine rule formula

Label the triangle so that each side has the small letter of the angle opposite it: side a is opposite angle A, side b opposite angle B and side c opposite angle C. Then:

The two versions say the same thing. Put the thing you want on top: sides on top to find a side, sines on top to find an angle. That saves a rearranging step. You only ever use two of the three fractions at a time.

When to use the sine rule

The sine rule needs a matching pair: a side and the angle opposite it, both known. Then you can find another side from its opposite angle, or another angle from its opposite side.

The quick test: if a side and its opposite angle are both given, use the sine rule. If not, you need the cosine rule. Many longer questions use both, one after the other, which is why they are taught together as the sine and cosine rules.

Using the sine rule to find a side

Example 1: two angles and a side

In triangle ABC, angle A = 40°, angle B = 65° and a = 8 cm. Find b.

a and A are a matching pair, and you want b, which is opposite B. Put the sides on top:

b = 11.3 cm to 3 significant figures. As a check, b is opposite the larger angle, so it should be longer than a. It is.

Example 2: find the third angle first

In triangle PQR, angle P = 35°, angle Q = 80° and the side opposite R is 15 cm. Find the side opposite P.

No side is paired with a known angle yet, so use the angle sum first: R = 180° − 35° − 80° = 65°. Now r = 15 and R = 65° are a matching pair.

The side opposite P is 9.49 cm.

Example 3: a distance you cannot measure

Two points A and B on a straight shore are 200 m apart. A boat C is out at sea, with angle CAB = 58° and angle CBA = 47°. How far is the boat from A?

The angle at the boat is 180° − 58° − 47° = 75°, and it is opposite the 200 m side, so that is the matching pair. AC is opposite the 47° angle at B.

The boat is 151 m from A. Surveyors used exactly this method, called triangulation, to map whole countries before satellites.

Try one now. Find the missing side or angle, then press Mark to check your answer in the course.

Using the sine rule to find an angle

Example 4: one answer

In triangle ABC, a = 9 cm, b = 7 cm and angle A = 70°. Find angle B.

a and A are the matching pair. Put the sines on top:

B = 47.0°. Here b is shorter than a, so B must be smaller than A, and only the acute answer works.

The ambiguous case of the sine rule

When you use the sine rule to find an angle, there can be two possible answers. That is because sin x = sin(180° − x): an acute angle and an obtuse angle share the same sine. Your calculator only ever gives the acute one.

Example 5: two possible triangles

In triangle ABC, a = 6 cm, b = 9 cm and angle A = 32°. Find angle B.

The calculator gives B = 52.6°. But B could also be 180° − 52.6° = 127.4°. Check whether each one fits in a triangle with A = 32°: 32° + 52.6° = 84.6° and 32° + 127.4° = 159.4°. Both leave room for a third angle, so there are two possible triangles: B = 52.6° or B = 127.4°.

Here x is the angle in degrees and y is its sine. The curve is symmetrical about 90°, so every value of sin between 0 and 1 is hit twice: once by an acute angle and once by its obtuse partner. The two marked angles both have a sine of 0.795.

A quick way to spot the ambiguous case: it can only happen when the angle you know is opposite the shorter of the two known sides. In Example 4, the known angle was opposite the longer side, 9 cm, so there was only one answer. In Example 5, it was opposite the shorter side, 6 cm, so there could be two.

Area of a triangle using sine

Often taught alongside the sine rule, this formula finds the area of any triangle from two sides and the angle between them:

Example 6: area from two sides and an angle

Two sides of a triangle are 8 cm and 11 cm, and the angle between them is 40°.

The area is 28.3 cm². The angle must be the one between the two sides you use. If you have the wrong angle, find the right one first, with the sine rule, the cosine rule or the angle sum.

Proof of the sine rule

Draw triangle ABC and drop a perpendicular of height h from C to the side AB. This makes two right-angled triangles.

In the left triangle, the hypotenuse is b, so h = b sin A. In the right triangle, the hypotenuse is a, so h = a sin B. Both expressions equal h:

Divide both sides by sin A sin B:

Dropping the perpendicular from a different corner brings in c/sin C in the same way.

Common mistakes

• Pairing the wrong side and angle. Each side must be opposite its angle. Label the triangle before you start.
• Using the sine rule without a matching pair. With two sides and the angle between them, you need the cosine rule.
• Missing the obtuse answer. When finding an angle, check whether 180° minus your answer also fits in the triangle.
• Calculator in radians. sin 40 in radian mode is 0.745, not 0.643. Check for D or DEG on the screen.
• Rounding too early. Keep the full value of sin B on your calculator before you press sin⁻¹.
• Using the wrong angle in ½ab sin C. C must be between sides a and b.

Is the sine rule on the formula sheet?

It depends on the exam. This was checked against the exam boards' sheets in November 2026.

The GCSE Higher sheet prints only the version with sides on top. You need to flip it, or rearrange, to find an angle yourself. In A-level Maths, the sine rule, cosine rule and area formula are assumed from GCSE, so learn them.

Check your answers in the Trigonometry Calculator, which solves any triangle and shows whether it used the sine rule or the cosine rule. The Trigonometry course builds from SOH CAH TOA to the sine rule, the cosine rule and the area of any triangle.

Practise the sine rule in a short timed paper

GCSE Maths Higher gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the whole course, so non-right-angled triangles come up regularly. In Scotland, the same practice opens as National 5 Maths.

Read next

Maths · TrigonometryCosine rule: formula, finding angles and worked examplesThe cosine rule for a side or an angle, when to use it instead of the sine rule, 6 worked examples, practice questions and what each exam gives you.