The party of a triangle can be found not only on perimeter and the area, but also on the set party and corners. Trigonometrical functions - a sine and a cosine are for this purpose used. Tasks meet their use in a school course of geometry and also in a high school course of analytical geometry and linear algebra.
Instruction
1. If one of the parties of a triangle and a corner between it and other its party is known, use trigonometrical functions - a sine and a cosine. Imagine a rectangular triangle of HBC at which the corner α is equal to 60 degrees. The triangle of HBC is shown in the drawing. As the sine, as we know, represents the relation of an opposite leg to a hypotenuse, and a cosine - the relation of an adjacent leg to a hypotenuse, for the solution of an objective use the following ratio between these parameters: α = NV/VSSootvetstvenno if you want to learn a leg of a rectangular triangle, express to sin it through a hypotenuse as follows: HB=BC*sin α
2. If in a statement of the problem, on the contrary, the triangle leg is given, find its hypotenuse, being guided by the following ratio between the set sizes: αПо analogies find BC=HB/sin the parties of a triangle and with use of a cosine, having changed the previous expression as follows: cos α = HC/BC
3. In elementary mathematics there is a concept of the theorem of sine. Being guided by the facts which are described by this theorem, it is also possible to find the parties of a triangle. In addition, it allows to find the parties of a triangle, inscribed in a circle if it is known the radius of the last is known. For this purpose use the ratio provided below: a/sin α=b/sin the theorem is applicable b=c/sin y=2REta when two parties and a corner of a triangle are known, or one of corners of a triangle and radius of the circle described around it is given.
4. Besides the theorem of sine, there is also similar to it in fact a theorem of cosines which, as well as previous, is also applicable to triangles of all three versions: rectangular, acute-angled and obtusangular. Being guided by the facts which prove this theorem, it is possible to find unknown sizes, using the following ratios between them: c^2=a^2+b^2-2ab*cos α