To find the parties of a triangle, it is necessary to know lengths of two parties and size of one corner. Or on the contrary - length of one party and size of two corners. The size of the third corner it is easy for 180 degrees to calculate the amounts of corners of a triangle from equality.
Instruction
1. Lengths of two parties of a triangle and size of a corner between them are known for two parties and a corner between nimiyesl, it is possible to find length of the third party having used the theorem of cosines: the square of length of the party of a triangle equals to the sum of squares of lengths of two of its other parties minus the doubled work of these parties on a cosine of the angle between them. From here we have: with = √ (and²+b²-2аb*cosC), where and and b – lengths of the known parties, With – the size of the corner concluded between these parties (opposite to the required party), with – length of the required party. Example 1. The triangle with the parties of 10 cm and 20 cm and a corner between them equal 60 degrees is given. To find length of the party. Decision. On the above-stated formula we receive: with = √ (10²+20²-2*10*20*cos60º)= √ (500-200)= √ 300 ~ 17.32 Answer: length of the party of a triangle, opposite to the parties lengths of 10 both 20 centimeters and size of a corner between them 60º - ~ 17.32 cm.
2. And storoneesl are known for two corners sizes of two corners and length of one parties of a triangle, it is the most convenient to find lengths of two other parties having used the theorem of sine: the relation of sine of corners of a triangle to lengths of the opposite parties are equal among themselves. sinA/a=sinB/b=sinC/с, where: a, b, c are lengths of the parties of a triangle, and A, B, C are sizes of opposite corners. What corners of a triangle are known – not important as, having used the fact that the sum of corners of a triangle is equal 180 degrees, it is possible to learn the size of an unknown corner easily. That is, for example, if sizes of corners And yes With and length of the party are known and, then length of the party with will be: with = a*sinC/sinA
3. If at the same basic data it is necessary to find length of the party of b, then to use the theorem of sine, it is necessary to know corner size In: as B=180º-A-C, length of the party of b it will be possible to find on a formula: b=a*sin(180º-Ç-æ)/sinAPrimer 2. Let in a triangle of ABC are known length of the party and =10 cm and sizes of corners of A=30 and C =20. To find length of the party of b. Decision: on the formula received above we receive: b=10*sin (180º-30º-20º)/sin30º=10*sin130º/0.5=5*sin130º ~ 3.83 Answer: length of the party of a triangle ~ 3.83 cm.