Trigonometry – the section of mathematics for studying the functions expressing various dependences of the parties of a rectangular triangle on sizes of acute angles at a hypotenuse. Such functions received naming trigonometrical, and for simplification of work with them trigonometrical identities were removed.
The concept of identity of mathematics means equality which is carried out at any values of arguments of the functions entering it. Trigonometrical identities are the equalities of trigonometrical functions proved and accepted for simplification of work with trigonometrical formulas. Trigonometrical function is an elementary function of dependence of one of legs of a rectangular triangle on the size of an acute angle at a hypotenuse. There are six main trigonometrical functions most often used: sin (sine), cos (cosine), tg (tangent), ctg (cotangent), sec (secant) and cosec (cosecant). These functions are called straight lines, there are also inverse functions, for example, a sine – an arcsine, a cosine – an arccosine, etc. Initially trigonometrical functions found reflection in geometry, then extended in other fields of science: physics, chemistry, geography, optics, probability theory and also acoustics, theory of music, phonetics, computer graphics and many others. It is now difficult to imagine mathematical calculations without these functions though far back in the past they were applied only in astronomy and architecture. Trigonometrical identities are applied to simplification of work with long trigonometrical formulas and their reduction to a digestible look. The main trigonometrical identities six, they are connected with direct trigonometrical functions: • tg? = sin? / cos?; • sin^2? + cos^2? = 1; • 1 + tg^2? = 1/cos^2?; • 1 + 1/tg^2? = 1/sin^2?; • sin (?/2-?) = cos?; • cos (?/2-?) = sin?. It is easy to prove these identities from properties of a ratio of the parties and corners in a rectangular triangle: sin? = BC/AC = b/c; cos? = AB/AC = a/c; tg? = b/a. The first identity of tg? = sin? / cos? follows from a ratio of the parties in a triangle and an exception of the party of c (hypotenuse) at division of sin into cos. In the same way the identity of ctg is defined? = cos? / sin?, as ctg? = 1/tg?. On a^2 Pythagorean theorem + b^2 = c^2. Let's divide this equality into c^2, we will receive the second identity: a^2/c^2 + b^2/c^2 = 1 => sin^2? + cos^2? = 1. Receives the third and fourth identities by division, respectively, into b^2 and a^2: a^2/b^2 + 1 = c^2/b^2 => tg^2? + 1 = 1/cos^2?; 1 + b^2/a^2 = c^2/a^2 => 1 + 1/tg^2? = 1/sin^? or 1 + ctg^2? = 1/sin^2?. The fifth and sixth main identities are proved through determination of the sum of acute angles of a rectangular triangle which is equal 90 ° or?/2. More difficult trigonometrical identities: formulas of addition of arguments, a double and threefold corner, decrease in degree, transformation of the sum or performing functions and also formulas of trigonometrical substitution, namely expression of the main trigonometrical functions through tg of a half corner: sin? = (2*tg?/2) / (1 + tg^2?/2); cos? = (1 – tg^2?/2) / (1 = tg^2?/2); tg? = (2*tg?/2) / (1 – tg^2?/2).