Two triangles labelled with the components of the law of sines. α, β and γ are the angles associated with the vertices at capital A, B, and C, respectively. Lower-case a, b, and c are the lengths of the sides opposite them. (a is opposite α, etc.)
where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle. When the last part of the equation is not used, the law is sometimes stated using the reciprocals;
The law of sines can be used to compute the remaining sides of a triangle when two angles and a side are known—a technique known as triangulation. It can also be used when two sides and one of the non-enclosed angles are known. In some such cases, the triangle is not uniquely determined by this data (called the ambiguous case) and the technique gives two possible values for the enclosed angle.
The law of sines is one of two trigonometric equations commonly applied to find lengths and angles in scalene triangles, with the other being the law of cosines.
The law of sines can be generalized to higher dimensions on surfaces with constant curvature.
Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in the 11th century contains the general law of sines. The plane law of sines was later stated in the 13th century by Nasīr al-Dīn al-Tūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles, and provided proofs for this law.
According to Glen Van Brummelen, "The Law of Sines is really Regiomontanus's foundation for his solutions of right-angled triangles in Book IV, and these solutions are in turn the bases for his solutions of general triangles." Regiomontanus was a 15th-century German mathematician.
The area T of any triangle can be written as one half of its base times its height. Selecting one side of the triangle as the base, the height of the triangle relative to that base is computed as the length of another side times the sine of the angle between the chosen side and the base. Thus depending on the selection of the base, the area of the triangle can be written as any of:
Multiplying these by 2/abc gives
The ambiguous case of triangle solution
When using the law of sines to find a side of a triangle, an ambiguous case occurs when two separate triangles can be constructed from the data provided (i.e., there are two different possible solutions to the triangle). In the case shown below they are triangles ABC and ABC′.
Given a general triangle, the following conditions would need to be fulfilled for the case to be ambiguous:
The only information known about the triangle is the angle α and the sides a and c.
The second equality above readily simplifies to Heron's formula for the area.
The sine rule can also be used in deriving the following formula for the triangle's area: Denoting the semi-sum of the angles' sines as , we have
where is the radius of the circumcircle: .
The spherical law of sines
The spherical law of sines deals with triangles on a sphere, whose sides are arcs of great circles.
Suppose the radius of the sphere is 1. Let a, b, and c be the lengths of the great-arcs that are the sides of the triangle. Because it is a unit sphere, a, b, and c are the angles at the center of the sphere subtended by those arcs, in radians. Let A, B, and C be the angles opposite those respective sides. These are dihedral angles between the planes of the three great circles.
Then the spherical law of sines says:
Consider a unit sphere with three unit vectors OA, OB and OC drawn from the origin to the vertices of the triangle. Thus the angles α, β, and γ are the angles a, b, and c, respectively. The arc BC subtends an angle of magnitude a at the centre. Introduce a Cartesian basis with OA along the z-axis and OB in the xz-plane making an angle c with the z-axis. The vector OC projects to ON in the xy-plane and the angle between ON and the x-axis is A. Therefore, the three vectors have components:
The scalar triple product, OA ⋅ (OB × OC) is the volume of the parallelepiped formed by the position vectors of the vertices of the spherical triangle OA, OB and OC. This volume is invariant to the specific coordinate system used to represent OA, OB and OC. The value of the scalar triple productOA ⋅ (OB × OC) is the 3 × 3 determinant with OA, OB and OC as its rows. With the z-axis along OA the square of this determinant is
Repeating this calculation with the z-axis along OB gives (sin c sin a sin B)2, while with the z-axis along OC it is (sin a sin b sin C)2. Equating these expressions and dividing throughout by (sin a sin b sin c)2 gives
where V is the volume of the parallelepiped formed by the position vector of the vertices of the spherical triangle. Consequently, the result follows.
It is easy to see how for small spherical triangles, when the radius of the sphere is much greater than the sides of the triangle, this formula becomes the planar formula at the limit, since
and the same for sin b and sin c.
Consider a unit sphere with:
Construct point and point such that
Construct point such that
It can therefore be seen that and
Notice that is the projection of on plane . Therefore
By basic trigonometry, we have:
Combining them we have:
By applying similar reasoning, we obtain the spherical law of sine:
For example, a tetrahedron has four triangular facets. The absolute value of the polar sine of the normal vectors to the three facets that share a vertex, divided by the area of the fourth facet will not depend upon the choice of the vertex:
^Sesiano just lists al-Wafa as a contributor. Sesiano, Jacques (2000) "Islamic mathematics" pp. 137–157, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN1-4020-0260-2