In mathematics there are several different approaches by means of which definitions of each of trigonometrical functions are given - through the solution of the differential equations, through ranks, the solution of the functional equations. There are also two options of geometrical interpretations of such functions, one of which defines them through ratios of the parties and acute angles in a rectangular triangle.
Instruction
1. Use basic definition of a sine of an acute angle in a triangle if from conditions it is known that it is a rectangular triangle and also lengths of its hypotenuse are given (C) and that leg (A) which lies opposite to the necessary corner (?). According to definition, the sine of this corner has to be equal to a ratio of length of the known leg to hypotenuse length: sin(?) = A/S.
2. If the triangle is rectangular, length of its hypotenuse is known (C), but also from legs there is only length (B) adjacent to that corner (?) which sine should be calculated, then in addition to definition from the previous step it is possible to involve also Pythagorean theorem. From it follows that length of an unknown leg is equal to a square root from the difference of the squared lengths of a hypotenuse and other leg. Substitute this expression in the formula received above: sin(?)=v (With? - In?) / Page.
3. Use Pythagorean theorem and in case in a rectangular triangle only lengths of both legs are known (And yes C). Hypotenuse length, according to the theorem, is equal to a square root from the sum of squares of lengths of legs. Replace with this expression hypotenuse length in a formula from the first step: sin(?) = And / v (And? + In?).
4. If lengths of the parties of a rectangular triangle are unknown, but the size of one of its acute angles (?) is given, then it is possible to calculate a sine of other acute angle (?) with use of tables of trigonometrical functions or the calculator. You proceed from the theorem of the sum of corners of a triangle in Euclidean geometry - she claims that this sum has to be always equal 180 °. As one of corners by definition is equal in a rectangular triangle 90 °, and another is given in statements of the problem, the size of the necessary corner will be equal 180 ° -90 °-?. Means to you it will be necessary only to calculate value of a sine of the angle: sin (90 °-?).
5. For calculation of value of a sine at the known size of a corner use, for example, the calculator which is built in the operating system of your computer. If it is Windows OS, then it is possible to start such application, having pressed a combination of the keys Ctrl + R, having entered the calc team, and then having clicked the OK button. For access to trigonometrical functions in the calculator switch it to the "engineering" or "scientific" mode - the corresponding point is in the section "Look" of the menu of this program.