Most often problems need to be solved with cosines in geometry. If this concept is used in other sciences, for example, in physics, then geometrical methods are applied. The theorem of cosines or a ratio in a rectangular triangle is usually applied.
It is required to you
- - knowledge of Pythagorean theorem, theorem of cosines;
- - trigonometrical identities;
- - calculator or Bradis's tables.
Instruction
1. By means of a cosine it is possible to find any of the parties of a rectangular triangle. For this purpose use a mathematical ratio in which to be said that a cosine of an acute angle of a triangle is the relation of an adjacent leg to a hypotenuse. Therefore, knowing an acute angle of a rectangular triangle, find its parties.
2. For example, the hypotenuse of a rectangular triangle is equal to 5 cm, and an acute angle at it 60º. Find a leg, adjacent to an acute angle. For this purpose use definition of a cosine of cos(α) = b/a where an is a hypotenuse of a rectangular triangle, b is the leg adjacent to a corner α. Then its length will be equal to b=a∙cos(α). Substitute b=5∙cos(60º) values = 5∙0,5=2,5 cm.
3. The third party with which is the second leg, find, having used c= Pythagorean theorem √ (5²-2.5²) ≈ 4.33 cm.
4. By means of the theorem of cosines it is possible to find the parties of triangles if two parties and a corner between them are known. To find the third party, find the sum of squares of two known parties, take away from it their doubled work increased by a cosine of the angle between them. Take a square root from the received result.
5. An example two parties are equal In a triangle to a=12 of cm, b=9 see the Corner between them makes 45º. Find the third party with. Apply the theorem of cosines of c= to finding of the third party √ (a²+b²-a∙b∙cos(α)). Having made substitution receive, c= √ (12²+9²-12∙9∙cos(45º)) ≈12.2 cm.
6. At the solution of tasks with cosines, use the identities allowing to come from this trigonometrical function to others and vice versa. Main trigonometrical identity: cos² (α) +sin² (α) =1; a ratio with a tangent and a cotangent: tg(α)=sin(α)/cos(α), ctg(α)=cos(α)/sin(α), etc. For finding of value of cosines of corners use the special calculator or Bradis's table.