Generally knowledge of length of one party and one corner of a triangle is not enough for determination of length of other party. These data can be enough for definition of the parties of a rectangular triangle and also an isosceles triangle. Generally a case it is necessary to know one more parameter of a triangle.
It is required to you
- Parties of a triangle, triangle corners
Instruction
1. For a start it is possible to consider special cases and to begin with a case of a rectangular triangle. If it is known that the triangle rectangular and is known one of its acute angles, then on length of one of the parties it is possible to find also the lrugy parties of a triangle. For finding of length of other parties it is necessary to know what party of a triangle is set - a hypotenuse or some of legs. The hypotenuse lies against a right angle, legs form a right angle. Consider a rectangular triangle of ABC with a right angle of ABC. Let its hypotenuse by AC and, for example, an acute angle of BAC be set. Then legs of a triangle will be equal: AB = AC*cos(BAC) (an adjacent leg to BAC corner), BC = AC*sin(BAC) (a leg, opposite to BAC corner).
2. Let the same corner of BAC and, for example, AB leg be set now. Then the hypotenuse of AC of this rectangular triangle is equal: AC = AB/cos(BAC) (respectively, AC = BC/sin(BAC)). Other leg of BC is on BC formula = AB*tg(BAC).
3. Other special case - if a triangle of ABC isosceles (AB = AC). Let BC basis be set. If BAC corner is set, then sides of AB and AC can be found on a formula: AB = AC = (BC/2)/sin(BAC/2). If the corner at foundation of ABC or ACB, then AB is set = by AC = (BC/2)/cos(ABC).
4. Let one of sides of AB or AC be set. If the corner of BAC, then BC = 2*AB*sin(BAC/2) is known. If the corner of ABC or a corner of ACB at the basis, then BC = 2*AB*cos is known (ABC).
5. Now it is possible to consider the general case of a triangle when length of one party and one corner is not enough for finding of length of other party. Let in a triangle of ABC the party of AB and one of corners, adjacent to it, for example, a corner of ABC be set. Then, knowing still the party of BC, according to the theorem of cosines it is possible to find the party of AC. It will be equal: AC = sqrt ((AB^2)+ (BC^2)-2*AB*BC*cos (ABC))
6. Let the party of AB and opposite be known to it ACB corner now. Let the corner of ABC be also known, for example. According to the theorem of sine of AB/sin (ACB) = AC/sin(ABC). Therefore, AC = AB*sin(ABC)/sin(ACB).