You will need
- - knowledge of the Pythagorean theorem, theorem of cosines;
- - trigonometric identities;
- calculator or tables Bradis.
With cosine, you can find any of the sides of a right triangle. To do this, use the mathematical relationship that States that the cosine of an acute angle triangle is the ratio of adjacent sides to the hypotenuse. Therefore, knowing the acute angle of a right triangle, find the side of it.
For example, the hypotenuse of a right triangle is 5 cm and the acute angle 60º with her. Locate adjacent to the acute corner side. To do this, use the definition of cosine cos(α)= b/a, where a is the hypotenuse of a right triangle, b is the side adjacent to the angle α. Then its length will be equal to b=a∙cos(α). Substitute values b=5∙cos(60º)= 5∙0,5=2,5 cm
The third side C, which is the second leg find using the Pythagorean theorem c=√(52-2,52)≈4.33 cm
Using the spherical law of cosines, you can find sides of triangles if you know two sides and the angle between them. To find the third side, find the sum of the squares of the two known sides, subtract from it twice the product multiplied by the cosine of the angle between them. From the result, extract the square root.
Example In triangle two sides are equal a=12 cm, b=9 cm Angle between them is 45º. Find the third side c. To find the third side use the cosine theorem c=√(a2+b2-a∙b∙cos(α)). Performing a lookup will get c=√(122+92-12∙9∙cos(45º))≈12,2 cm
When solving problems with cosines, use of identity, allowing to pass away from this trigonometric function to another, and Vice versa. The basic trigonometric identity: cos2(α)+sin2(α)=1; the ratio of tangent and cotangent: tg(α)=sin(α)/cos(α), ctg(α)=cos(α)/sin(α), etc. To find the value of the cosines of the angles use a special calculator or table Bradis.