For sizes of corners which lie in triangle tops and also the parties forming them, certain ratios are characteristic. Usually they say by means of trigonometrical functions - through a cosine and a sine. If length of each of the parties of a triangle is given, then it is possible to remove also sizes of its corners.
Instruction
1. Use the theorem of cosines to calculate sizes of any corner of any triangle with the parties of A, B and C. According to it the square of length of one of the parties equals to the sum of squares of lengths of other parties from which the work of these lengths on a cosine lying in vertex of angle α is subtracted. Thus, the cosine is expressed through the following formula: cos(α) = (C²-A²+B²) / (A*B*2). To obtaining size of this corner in degrees it is necessary to apply inverse function to the received expression: α = arccos ((C²-A²+B²) / (A*B*2)). So you will be able to calculate the size of the corner lying opposite to the party And.
2. Calculate two remained corners, having used the same formula, having substituted in it values of lengths of the known parties. However for receiving simpler expression without a large number of mathematical calculations it is necessary to take in attention other postulate from trigonometry, namely the theorem of sine. According to it the relation of length of one of the parties to a sine of an opposite corner allows to remove other corners. It means that a sine of one of corners, for example, β, B lying opposite to the relevant party, it is possible to express through value of length of the party C and known corners α.
3. Execute multiplication of length of B by a sine of the angle α, having divided result into length of C. Thus, sin(β) = sin(α)/C*B*. The size of this corner is calculated in degrees by means of inverse function of an arcsine which looks as follows: β = arcsin (sin(α)/C*B).
4. Remove the size of the last corner γ through any of the formulas received earlier, having substituted the corresponding lengths of the parties. Easier way consists in use of the theorem of the sum of corners of a triangle. It is known that this sum always is 180 °. As two corners are known already, their sum needs just to be subtracted from 180 ° that the size of the last turned out: γ = 180 ° - (α+β).