Knowledge of value of a cosine of the angle allows to find the size of this corner in top of any triangle. But it is impossible to recognize length of the party of such figure by the only parameter, any additional related sizes are necessary. If they are given in conditions, the choice of a formula of calculation will depend on what parameters are chosen as addition to a cosine of the angle.
Instruction
1. If except value of a cosine of the angle lengths of couple of parties are known (b and c) which form this corner, for calculation of size of the unknown party (a) can use the theorem of cosines. She claims that the square of length of the necessary party will be equal to the sum of squares of lengths of two others if to reduce it by the doubled work of lengths of the same parties on the cosine of the angle, known from conditions, between them: a² = b² + with² - 2*a*b*cos(α).
2. As corner size α is unknown to you and to calculate it there is no need, designate the variable (cosine of the angle) given in conditions by any letter (for example, f) also substitute in a formula: a² = b² + with² - 2*a*b*f. Get rid of degree in the left part of expression to receive a final formula of calculation of length of the required party in a general view: a = √ (b²+c²-2*a*b*f).
3. To find length of the party (a) provided that except value of a cosine (f = cos(α)) lying opposite to this side of angle is given the size of other corner (β) and length of the party (b) lying opposite to it, it is possible to use the theorem of sine. According to it the relation of required length to a sine of an opposite corner is equal to the relation of length of the known party to a sine of the angle which is given in conditions too: a/sin(a) = b/sin(β).
4. The sum of squares of a sine and cosine of the same corner is equal to unit - use this identity to express a sine in the left part of equality through the cosine set in conditions: a/√ (1-f²) = b/sin(β). Make a formula of calculation of length of the necessary party in a general view, having transferred a fraction denominator from the left part of identity to right: a = √ (1-f²)*b/sin(β).
5. In a rectangular triangle for calculation of sizes of the parties it is enough to add a cosine of an acute angle (f = cos(α)) one parameter - length any of the parties. To find length of the leg (b) adjoining top which cosine of the angle is known increase this size by length of a hypotenuse (c): b = f*c. If it is necessary to calculate hypotenuse length, and length of a leg is known, transform this formula as appropriate: c = b/f.