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Concept# Law of sines

Summary

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.
H.J.J. Wilson'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. 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.
According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to Abu-Mahmud Khojandi, Abu al-Wafa' Buzjani, Nasir al-Din al-Tusi and Abu Nasr Mansur.
Ibn Muʿādh al-Jayyānī's The book of unknown arcs of a sphere in the 11th century contains the spherical law of sines. The plane law of sines was later stated in the 13th century by Nasīr al-Dīn al-Tūsī.

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Trigonometry () is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.

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In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and (see Fig. 1), the law of cosines states: The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then and the law of cosines reduces to The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given.

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