law of sines
- Also called:
- sine rule
- Related Topics:
- sine
What does the law of sines state?
What is the spherical law of sines?
Who were some early contributors to the law of sines?
How did the law of sines develop during the Renaissance?
law of sines, principle of trigonometry stating that the lengths of the sides of any triangle are proportional to the sines of the opposite angles. That is, when a, b, and c are the sides and A, B, and C are the opposite angles.
History
Variant: Spherical law of sines
In spherical trigonometry, which deals with triangles on the surface of a sphere, the law of sines takes a modified form:.Here, a, b, and c represent the lengths of the sides of a spherical triangle, measured as the central angles subtended by the arcs at the center of the sphere, and A, B, and C are the corresponding opposite angles.
An early form of the law of sines appears in the work of the 2nd-century astronomer Ptolemy, who related chord lengths to angles in his studies. Further developments came from scholars of the medieval Islamic world—the 13th-century Persian polymath Naṣīr al-Dīn al-Ṭūsī provided a general proof of the law of sines for plane triangles, stating that in any triangle the sides are proportional to the sines of their opposite angles.
During the Renaissance, the law of sines was systematically presented in German astronomer-mathematician Regiomontanus’s De triangulis omnimodis libri quinque (1533; “Five Books on Triangles of Every Kind”), one of the earliest comprehensive works on trigonometry in Europe. His treatment of both plane and spherical triangles, including geometric derivations of the law, helped establish it as a standard method for solving triangles and supported its growing use in astronomy and mathematical computation.
