# 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

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 the opposite side

*c*that cuts the original triangle into two right triangles, and call the length of this line

*h*. Therefore:

*A*and side

*a*will yield:

## See also