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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.
There are several equivalent ways for defining trigonometric functions, and the proofs of the trigonometric identities between them depend on the chosen definition. The oldest and most elementary definitions are based on the geometry of right triangles and the ratio between their sides.
The spherical law of sines is given by the formula ... The proof (Todhunter, [1] Art.49) of the first formula starts from the identity = ...
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The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x .
The formula is credited to Heron (or Hero) of Alexandria (fl. 60 AD), [3] and a proof can be found in his book Metrica. Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier, [ 4 ] and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible ...
Fig. 7a – Proof of the law of cosines for acute angle γ by "cutting and pasting". Fig. 7b – Proof of the law of cosines for obtuse angle γ by "cutting and pasting". One can also prove the law of cosines by calculating areas. The change of sign as the angle γ becomes obtuse makes a case distinction necessary. Recall that
The law of sines (also known as the "sine rule") for an arbitrary triangle states: [85] = = = =, where is the area of the triangle and R is the radius of the circumscribed circle of the triangle: