# Online Encyclopedia

# Law of sines

In trigonometry, the **law of sines** (or **sine law**) is a statement about arbitrary triangles in the plane. If the sides of the triangle are (lower-case) *a*, *b* and *c* and the angles opposite those sides are (capital) *A*, *B* and *C*, then the law of sines states

This formula is useful to compute the remaining sides of a triangle if two angles and a side is known, a common problem in the technique of triangulation. It can also be used when two sides and one of the non-enclosed angles are known; in this case, the formula may give two possible values for the enclosed angle. When this happens, often only one result will cause all angles to be less than 180°; in other cases, there are two valid solutions to the triangle.

The reciprocal of the number described by the sine law (i.e. *a*/sin(*A*)) is equal to the diameter *D* of the triangle's circumcircle (the unique circle through the three points *A*, *B* and *C*). The law can therefore be written

## Derivation

Make a triangle with sides *a*, *b*, and *c*, and opposite angles *A*, *B*, and *C*. Make a line from the angle *C* to its opposite side *c* that cuts the figure into two right triangles, and call the length of this line *h*.

It can be observed that:

- and

Therefore:

and

Doing the same thing with the line drawn between angle *A* and side *a* will yield: