Cosine theorem . In Geometry and more specifically in Euclidean geometry , it is a trigonometry theorem that in each triangle indicates that the square of the length of each side is related to the squares of the remaining sides and the angles that they comprise.
Summary
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- 1
- 2 Implications and importance.
- 3 See also.
- 4
Definition.
Let any triangle be like the one in the figure:
The cosine theorem states that the square of the length of one side is the sum of the squares of the lengths of the remaining sides minus the double of the product of those lengths multiplied by the cosine of the angle opposite the side in question.
This is expressed through the expressions:
Implications and importance.
The cosine theorem plays a very important role in the calculation and definition of surfaces and volumes, along with the triangular inequality, the cosine and tangent theorem allow a characterization of the constituent elements (the lengths of the sides and amplitudes of the interior angles of the triangles), so that it can first be distinguished that they make up a triangle and also their relationship with their angles, among many other properties.
It is considered a generalization of the Pythagorean Theorem that applies only to right triangles, while the cosine theorem establishes a characterization on any type of triangles, including, as a particular case, the rectangles, for which, the cosine of the angle opposite to the hypotenuse is 0, returning to the historical result.
For example, let’s look at the case of the particularization of the isosceles according to the theorem:
where l is the length of the equal side and b the length of the base side.
