Irrational winding of a torus
Template:Trigonometry In trigonometry, the law of cotangents[1] relates the radius of the inscribed circle of a triangle (the inradius) to its sides and angles.
Statement
Using the usual notations for a triangle (see the figure at the upper right), where are the lengths of the three sides, are the angles opposite those three respective sides, is the semi-perimeter, that is, , and is the radius of the inscribed circle, the law of cotangents states that:
And furthermore that the inradius is given by:
Proof
In the upper figure, the points of tangency of the incircle with the sides of the triangle break the perimeter into 6 segments, in 3 pairs. In each pair the segments are of equal length. For example, the 2 segments adjacent to vertex A are equal. If we pick one segment from each pair, their sum will be the semiperimeter s. An example of this is the segments shown in color in the figure. The two segments making up the red line add up to a, so the blue segment must be of length s−a. Obviously, the other 5 segments must also have lengths s−a, s−b, or s−c, as shown in the lower figure.
By inspection of the figure, using the definition of the cotangent function, we have:
and similarly for the other two.
To get the inradius, note that the area of the triangle is also divided into 6 smaller triangles, also in 3 pairs, with the triangles in each pair having the same area. For example, the two triangles near vertex A, being right triangles of width s−a and height r, each have an area of . So those two triangles together have an area of , and the area of the whole triangle is therefore:
But, by Heron's formula, the area is also
so
Some proofs using the Law of cotangents
A number of other results can be derived from the law of cotangents.
- First Mollweide's formula. From the addition formula and the Law of cotangents we have:
- Second Mollweide's formula. From the addition formula and the Law of cotangents we have:
- .
- Here, an extra step is required to transform a product into a sum, according to the sum/product formula.
- This gives the result: as required.
- The Law of tangents can also be derived from this Template:Harv, as can the Pythagorean theorem.
See also
References
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- ↑ The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960.