The Law of Sines, Including the Ambiguous Case

The formula and the labelling convention

Label a triangle with angles $A$, $B$ and $C$, and call the side opposite each angle $a$, $b$ and $c$ respectively. The side opposite an angle is the one that does not touch it. With that convention the law of sines states:

Here $R$ is the radius of the circle that passes through all three vertices, called the circumscribed circle. You only ever use two of the three fractions at once, so the law is really a proportion between a side, its opposite angle, another side and that side's opposite angle. It can also be written upside down, $\frac{\sin A}{a} = \frac{\sin B}{b}$, which is more convenient when the unknown is an angle.

The law works for every triangle, not only right triangles, which is why it belongs to what is often called oblique triangle trigonometry. In a right triangle with $C = 90^\circ$ it reduces to the familiar $\sin A = \frac{a}{c}$, because $\sin 90^\circ = 1$.

Why the law of sines is true

Drop a perpendicular from vertex $C$ to side $c$ and call its length $h$. This splits the triangle into two right triangles. In the one containing angle $A$, the hypotenuse is $b$, so $h = b\sin A$. In the one containing angle $B$, the hypotenuse is $a$, so $h = a\sin B$.

Both expressions equal the same height, so $b\sin A = a\sin B$. Dividing both sides by $\sin A\sin B$ gives $\frac{a}{\sin A} = \frac{b}{\sin B}$. Repeating the argument with the altitude from a different vertex brings in side $c$. If angle $A$ is obtuse the foot of the altitude lands outside the triangle, but the same equations hold because $\sin(180^\circ - A) = \sin A$.

The same altitude gives the area formula. The area is half the base times the height, so with base $c$ and height $b\sin A$:

For example, a triangle with sides 8 and 11 meeting at a $50^\circ$ angle has area $\frac{1}{2}(8)(11)\sin 50^\circ \approx 33.71$ square units.

When to use it: AAS and ASA

The law of sines needs at least one complete pair: a side together with the angle opposite it. If you know two angles, you know the third, because the angles of a triangle add to $180^\circ$. So any problem that gives two angles and one side can be completed with the law of sines. Those cases are called AAS (angle, angle, side) and ASA (angle, side, angle).

Use the given side in every calculation rather than a side you computed and rounded. That keeps rounding errors from building up.

The ambiguous case (SSA)

When you know two sides and an angle opposite one of them, the information may describe no triangle, exactly one, or two different triangles. The reason is that the sine of an acute angle and the sine of its supplement are equal, so solving $\sin B = k$ gives two candidate angles, $B$ and $180^\circ - B$, and both may fit.

Suppose angle $A$ is acute, with $a$ the side opposite it and $b$ the other known side. Compute the height $h = b\sin A$. Then:

• If $a < h$, the side is too short to reach the base: no triangle.
• If $a = h$, it just reaches: one right triangle.
• If $h < a < b$, it can swing to two positions: two triangles.
• If $a \ge b$, only one position works: one triangle.

If $A$ is obtuse, there is one triangle when $a > b$ and none otherwise, because the side opposite the obtuse angle must be the longest.

A surveying problem

Surveyors use the law of sines to measure distances they cannot walk. Two points $P$ and $Q$ on one bank of a river are 100 m apart. A tree $T$ stands on the far bank. The angle at $P$ between the bank and the line of sight to the tree is $52^\circ$, and at $Q$ it is $71^\circ$.

The same triangulation idea, repeated over large networks of triangles, was used to map entire countries before satellite positioning existed, and it is still the basis of how GPS receivers and phone networks estimate position from angles and distances.

Law of sines or law of cosines?

Look at what you are given. If you have a side and its opposite angle, plus one more piece, use the law of sines. If you have two sides and the angle between them (SAS), or all three sides (SSS), no complete pair is available, so start with the law of cosines. After one application of the law of cosines you will usually have a pair, and the law of sines can finish the job, although it is safer to find the largest angle first with the law of cosines to avoid the ambiguity of the inverse sine.

The full decision process, case by case, is set out in our guide to solving triangles. For the angle values themselves, the unit circle explains why $\sin\theta = \sin(180^\circ - \theta)$, which is the source of the ambiguous case.

Common mistakes

• Pairing a side with an angle that is not opposite it. Draw and label the triangle first.
• Forgetting the second solution $180^\circ - B$ in the SSA case.
• Accepting a second solution that makes the angles add to more than $180^\circ$.
• Using a calculator in radian mode with angles in degrees.
• Rounding intermediate angles heavily, which can shift the final sides noticeably.

A quick plausibility test catches most errors: the longest side must face the largest angle and the shortest side the smallest angle. The law of cosines calculator solves any triangle from three known parts and reports both solutions in the ambiguous case, so you can check your work.

Frequently asked questions