Sinus theorem . In Geometry and more specifically in Euclidean geometry , it is a trigonometry theorem that in each triangle indicates that there is a constant ratio between each side and the sine of the corresponding interior angle. And as a more significant detail, the value of said proportion is the diameter of the circumference circumscribed to the triangle in question.
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:
To the proportional relationship:
where R is the radius of the circumference circumscribed to the triangle, it is known by the sine theorem .
Implications and importance.
The sine theorem adds a new property to the calculation and definition, along with the triangular inequality, the cosine and tangent theorem that characterizes its 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 the relationship between them with their angles and the radius of the circumscribed circle centered on the common point of the perpendicular bisectors.
In the case of the equilateral triangle the theorem is simply reduced to the line:
where a is the length of any of its sides and h , the height corresponding to it.
If it were an isosceles triangle , the expression changes to:
Now with right triangles, assuming c is the hypotenuse, the sine theorem is written:
That they are the well-known definitions of the trigonometric functions and also the result known as Thales’ Theorem is perceived in the final equality
