This article is about the law of sines in trigonometry. For the law of sines in physics, see Snell's law.
Law of Sines
Figure 1, With circumcircle
Figure 2, Without circumcircle
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.)
Trigonometry
Outline
History
Usage
Functions (sin, cos, tan, inverse)
Generalized trigonometry
Reference
Identities
Exact constants
Tables
Unit circle
Laws and theorems
Sines
Cosines
Tangents
Cotangents
Pythagorean theorem
Calculus
Trigonometric substitution
Integrals (inverse functions)
Derivatives
Trigonometric series
Mathematicians
Hipparchus
Ptolemy
Brahmagupta
al-Hasib
al-Battani
Regiomontanus
Viète
de Moivre
Euler
Fourier
v
t
e
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]
side of a triangle if two sides and an angle opposite to one of them is known (this side can also be found by two applications of the lawofsines): a...
+\beta )}}.} The lawof tangents, although not as commonly known as the lawofsines or the lawof cosines, is equivalent to the lawofsines, and can be used...
do Gaio was unearthed in 1966, and is on display in the Museu de Sines (English: Sines Museum). Discovered in May 1966, the 'treasure' was unearthed by...
right ones and using the above definition ofsine. The lawofsines is useful for computing the lengths of the unknown sides in a triangle if two angles...
opposite the respective angles (as shown in the diagram). The lawofsines (also known as the "sine rule") for an arbitrary triangle states: a sin A = b sin...
functions Lawofsines List of periodic functions List of trigonometric identities Madhava series Madhava's sine table Optical sine theorem Polar sine—a generalization...
expressed by the lawofsines are equal to the diameter of the circumscribed circle of the triangle (or to its reciprocal, depending on how the law is expressed)...
relations: the lawof cotangents and Mollweide's formula. To find an unknown angle, the lawof cosines is safer than the lawofsines. The reason is that...
yielding the result stated by the theorem. In the above diagram, use the lawofsines on triangles △ABD and △ACD: Angles ∠ ADB and ∠ ADC form a linear pair...
the lengths of chords are applications of the lawofsines. And Archimedes' theorem on broken chords is equivalent to formulas for sinesof sums and differences...
tetrahedra is 5-dimensional. See: Lawofsines The lawof cosines for the tetrahedron relates the areas of each face of the tetrahedron and the dihedral...
and compositional inverse. The lawofsines, or sine rule, states that the ratio of the length of a side to the sineof its corresponding opposite angle...
{\displaystyle (\cos a-\cos b)^{2}\approx 0} etc.; see Spherical lawof cosines.) The spherical lawofsines is given by the formula sin A sin a = sin B sin...
and c {\displaystyle c} are the lengths of the sides of the triangle, or equivalently (using the lawofsines) by sin A : sin B : sin C {\displaystyle...
the lawofsines is attributed; he wrote "The Book of Unknown Arcs of a Sphere" in the 11th century. This formula relates the lengths of the sides of any...
Arcsine law may refer to: Arcsine distribution Arcsine laws (Wiener process), describing one-dimensional random walks Erdős arcsine law, concerning the...
readily read the missing term of the lawofsines or the roots of the quadratic and cubic equation. Nomogram for the lawofsines Nomogram for solving the...
Morrie's law Proofs of trigonometric identities Pythagorean trigonometric identity Tangent half-angle formula Solution of triangles LawofsinesLawof cosines...
from the lawofsines. The inradius is r = ( s − a ) ( s − b ) ( s − c ) s . {\displaystyle r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}}.} The lawof cotangents...
Babylon 5, Martial Law, Diagnosis: Murder, That's My Bush!, Star Trek: Enterprise, ChuckleVision, Family Tree and The Crown. Siner is one of ten actors to...
{OP_{3}}}|}{2\sin \theta }},} where θ is the angle ∠P1P2P3. This formula uses the lawofsines. If the three points are given by their coordinates (x1,y1), (x2,y2)...
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