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.)
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
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.[1]
H. J. J. Winter's book Eastern Science states that the 7th century Indian mathematician Brahmagupta describes what we now know as the law of sines in his astronomical treatise Brāhmasphuṭasiddhānta.[2] In his partial translation of this work, Colebrooke translates Brahmagupta's statement of the sine rule as: The product of the two sides of a triangle, divided by twice the perpendicular, is the central line; and the double of this is the diameter of the central line.[3]
Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in the 11th century contains the spherical law of sines.[5] 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.[6]
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."[7] Regiomontanus was a 15th-century German mathematician.
With the side of length a as the base, the triangle's altitude can be computed as b sin γ or as c sin β. Equating these two expressions gives
and similar equations arise by choosing the side of length b or the side of length c as the base of the triangle.
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 side a is shorter than the side c (i.e., a < c).
The side a is longer than the altitude h from angle β, where h = c sin α (i.e., a > h).
If all the above conditions are true, then each of angles β and β′ produces a valid triangle, meaning that both of the following are true:
From there we can find the corresponding β and b or β′ and b′ if required, where b is the side bounded by vertices A and C and b′ is bounded by A and C′.
If the lengths of two sides of the triangle a and b are equal to x, the third side has length c, and the angles opposite the sides of lengths a, b, and c are α, β, and γ respectively then
In the identity
the common value of the three fractions is actually the diameter of the triangle's circumcircle. This result dates back to Ptolemy.[8][9]
Deriving the ratio of the sine law equal to the circumscribing diameter. Note that triangle ADB passes through the center of the circumscribing circle with diameter d.
As shown in the figure, let there be a circle with inscribed and another inscribed that passes through the circle's center O. The has a central angle of and thus , by Thales's theorem. Since is a right triangle,
where is the radius of the circumscribing circle of the triangle.[9] Angles and lie on the same circle and subtend the same chordc; thus, by the inscribed angle theorem, . Therefore,
Rearranging yields
Repeating the process of creating with other points gives
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.
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.
A purely algebraic proof can be constructed from the spherical law of cosines. From the identity and the explicit expression for from the spherical law of cosines
Since the right hand side is invariant under a cyclic permutation of the spherical sine rule follows immediately.
The figure used in the Geometric proof above is used by and also provided in Banerjee[12] (see Figure 3 in this paper) to derive the sine law using elementary linear algebra and projection matrices.
Define a generalized sine function, depending also on a real parameter :
The law of sines in constant curvature reads as[1]
By substituting , , and , one obtains respectively , , and , that is, the Euclidean, spherical, and hyperbolic cases of the law of sines described above.[1]
Let indicate the circumference of a circle of radius in a space of constant curvature . Then . Therefore, the law of sines can also be expressed as:
More generally, for an n-dimensional simplex (i.e., triangle (n = 2), tetrahedron (n = 3), pentatope (n = 4), etc.) in n-dimensional Euclidean space, the absolute value of the polar sine of the normal vectors of the facets that meet at a vertex, divided by the hyperarea of the facet opposite the vertex is independent of the choice of the vertex. Writing V for the hypervolume of the n-dimensional simplex and P for the product of the hyperareas of its (n − 1)-dimensional facets, the common ratio is
^Sesiano just lists al-Wafa as a contributor. Sesiano, Jacques (2000). "Islamic mathematics". In Selin, Helaine; D'Ambrosio, Ubiratan (eds.). Mathematics Across Cultures: The History of Non-western Mathematics. Springer. pp. 137–157. ISBN1-4020-0260-2.
^Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN978-0-691-11485-9.