Generally, i.e. when there is no data on whether the triangle is equilateral, isosceles, rectangular, it is necessary to use trigonometrical functions for calculation of lengths of its parties. Rules of their application are defined by theorems which and are called - the theorem of sine, cosines and tangents.
Instruction
1. One of ways of calculation of lengths of the parties of any triangle assumes use of the theorem of sine. According to it ratios of lengths of the parties and sine opposite corners of a triangle are equal to them. It allows to remove a formula of length of the party for those cases when from statements of the problem there is at least one known party and two corners in figure tops. If any of these two corners (α and β) does not lie between the known party And yes calculated In, then increase length of the known party by a sine of the known corner adjoining to it β and divide into a sine of other known corner and: In = А*sin(β)/sin(α).
2. If one (γ) (α and γ) the known corners is formed of two by the parties, length of one of which (A) is given in conditions, and the second (C) is required to be found, apply the same theorem. The decision can be consolidated to the formula received in the previous step if to remember also the theorem of the sum of corners in a triangle - this size is always equal 180 °. In a formula the corner β which according to this theorem can be calculated if to take away from 180 ° sizes of two known corners is unknown. Substitute this value in equality, and you receive formula B = A*sin (180 °-α-γ) / sin(α).
3. For initial conditions in which lengths of two parties are given (And yes C) and corner size between them (γ), the theorem of cosines allowing to find length of the third party approaches (C). For this purpose square lengths of the known parties and put, and from the received result take away their doubled work increased by a cosine of the angle of the known size. It is necessary to take a square root from the number received in such a way: With = √ (And² + In²-2*А*В*cos(γ)).
4. If the triangle can be entered in a circle of the known radius of R, and in conditions the corner size α, lying opposite to the party And which length should be calculated is given, then the formula will be rather simple. Find the work of radius on a sine of the known corner and double result: And = 2*R*sin(α).