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Terms with infinitely many sine factors would necessarily be equal to zero. When only finitely many of the angles are nonzero then only finitely many of the terms on the right side are nonzero because all but finitely many sine factors vanish. Furthermore, in each term all but finitely many of the cosine factors are unity.
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.
For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six ...
Trigonometry has been noted for its many identities, that is, equations that are true for all possible inputs. [83] Identities involving only angles are known as trigonometric identities. Other equations, known as triangle identities, [84] relate both the sides and angles of a given triangle.
Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jacob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Vieta. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer.
Some infinite sets truly have more elements than others in a deep mathematical way, and Cantor proved it. There is the first infinite size, the smallest infinity, which gets denoted ℵ₀. That ...
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