# Regulation of cosines with difficult triangulars

Exists any kind of mathematical value to the reality that the regulation of cosines ...

$$ \cos(\textrm{angle between }a\textrm{ and }b) = \frac{a^2 + b^2 - c^2}{2ab} $$

... for a difficult triangular generates a cosine $< -1$ (when $c > a+b$), or $> 1$ (when $c < \left|a-b\right|$)

As an example, $a = 3$, $b = 4$, $c = 8$ returns $\cos(\textrm{angle }ab) = -39/24$.

Or $a = 3$, $b = 5$, $c = 1$ returns $\cos(\textrm{angle }ab) = 33/30$.

Something to do with hyperbolic geometry/cosines?

This is not a straight an issue of hyperbolic geometry yet of *intricate Euclidean geometry *. The building and construction of "difficult" triangulars coincides as the building and construction of square origins of adverse numbers, when taking into consideration the works with the vertices of those triangulars have to have. If you compute the works with of a triangular with sides 1,3,5 or 3,4,8 you get intricate numbers. In average actual - coordinate Euclidean geometry this suggests there is no such triangular. If intricate works with are allowed, the triangular exists, yet not all its factors show up in illustrations that just stand for the actual factors.

In aircraft analytic geometry where the Cartesian works with are permitted to be intricate, principles of factor, line, circle, made even range, dot - item, and also (with ideal interpretations) angle and also cosine can be analyzed making use of the very same solutions. This semiotics expands the noticeable (actual - coordinate) Euclidean geometry to one where any kind of 2 circles converge, yet perhaps at factors with intricate works with. We "see" just the part of factors with actual works with, yet the building and construction that constructs a triangular with offered ranges in between the sides remains to function efficiently, and also some solutions of the regulation of Cosines will certainly remain to hold.

There are absolutely relationships of this image to hyperbolic geometry. One is that $cos(z)=cosh(iz)$ so you can see the hyperbolic cosine and also cosine as the very same as soon as intricate works with are allowed. An additional is that the Pythagorean statistics on the facility aircraft, taken into consideration as a 4 - dimensional actual room, is of the kind $x^2 + y^2 - w^2 - u^2$, to make sure that the locus of facility factors at range $0$ from the beginning has duplicates of the hyperboloid version of hyperbolic geometry. Yet there is no embedding of the hyperbolic aircraft as a straight subspace of the facility Euclidean aircraft, so we do not obtain from this a less complicated means of thinking of hyperbolic geometry.

To aid envision what is taking place it is brightening to compute the works with of a triangular with sides 3,4,8 or various other difficult instance, and also the dot - items of the vectors entailed.

For some a, b, and also c that create a triangular : raising the size of c raises the action of angle C and also as m ∠ C comes close to 180 °, cos C comes close to -1 ; lowering the size of c raises the action of angle C and also as m ∠ C comes close to 0 °, cos C comes close to 1. Expanding this pattern, it makes good sense that if c > a + b, c has actually grown previous making a triangular with an and also b, so cos C < -1, and also if c < |a-b |, c has actually obtained smaller sized previous making a triangular with an and also b, so cos C > 1.

In hyperbolic geometry, the definition of lines (and also therefore triangulars) is various and also the amount of the actions of the angles in a triangular is much less than 180 °. There is a Hyperbolic Law of Cosines, yet it's not fairly the very same.

I do not assume there's a reasonable means to connect hyperbolic cosine to the Law of Cosines in Euclidean geometry.

One can confirm the triangular inequality in any kind of abstract internal item room, such as a Hilbert room ; it issues of the Cauchy-Schwarz inequality $\langle a, b \rangle^2 \le ||a||^2 ||b||^2$. In order for the triangular inequality to fall short, the Cauchy-Schwarz inequality needs to fall short, and also in order for Cauchy-Schwarz to fall short (which represents the "cosine" no more being in between $1$ and also $-1$), some axiom of an abstract internal item room needs to be surrendered. One selection is to surrender positive-definiteness ; to put it simply, sizes of vectors are no more constantly non-negative. This brings about geometries like Minkowski spacetime which pertain to relativity. In Minkowski spacetime, there is a *reverse * triangular inequality rather.

Modify : I need to additionally state that the "device round" in Minkowski spacetime is a version for hyperbolic geometry called the hyperboloid model.