Laws of trigonometry 2 -- Sage

We retake the example of Laws of trigonometry 1 but with a different lexical ordering.

R = PolynomialRing (QQ, 'a,A,b,B,c,C', order='lex') a,A,b,B,c,C = R.gens() coslawa = (1+A^2)*(b^2+c^2-a^2) - 2*b*c*(1-A^2) coslawb = (1+B^2)*(c^2+a^2-b^2) - 2*c*a*(1-B^2) coslawc = (1+C^2)*(a^2+b^2-c^2) - 2*a*b*(1-C^2) Icos = Ideal (coslawa,coslawb,coslawc) Gcos = Icos.groebner_basis('libsingular:std') len (Gcos)

$\newcommand{\Bold}[1]{\mathbf{#1}}34$

Gcos[-1].factor()

$\newcommand{\Bold}[1]{\mathbf{#1}}c \cdot (- b + c) \cdot (- b B^{2} C - b C + B c C^{2} + B c) \cdot (b B^{2} C + b C + B c C^{2} + B c)$

And voilà, we immediately find the result we were looking for: the law of sines and an expression that looks very much like it. I.e., $\frac{\sin \beta}b = \frac{\sin \gamma}c$ or $\frac{\sin \beta}b = - \frac{\sin \gamma}c$.

The additional rule is not very surprising, because the cosine laws cannot distinguish between an angle and its negative as their cosines remain the same.

Finally, we compute a Gröbner basis using total degree ordering degrevlex (the default)

R = PolynomialRing (QQ, 'a,A,b,B,c,C', order='degrevlex') a,A,b,B,c,C = R.gens()

Note that the monomial ordering is indeed different:

(a+b+1)^2

$\newcommand{\Bold}[1]{\mathbf{#1}}a^{2} + 2 a b + b^{2} + 2 a + 2 b + 1$

The higher degree terms are now in front.

The resulting Gröbner basis is now shorter than in the lexicographical case:

coslawa = (1+A^2)*(b^2+c^2-a^2) - 2*b*c*(1-A^2) coslawb = (1+B^2)*(c^2+a^2-b^2) - 2*c*a*(1-B^2) coslawc = (1+C^2)*(a^2+b^2-c^2) - 2*a*b*(1-C^2) Icos = Ideal (coslawa,coslawb,coslawc) Gcos = Icos.groebner_basis('libsingular:std') len (Gcos)

$\newcommand{\Bold}[1]{\mathbf{#1}}20$

The resulting basis is now ordered by degree.

[f.degree() for f in Gcos]

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[9, 9, 8, 8, 8, 7, 7, 7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 4, 4, 4\right]$

The last three basis elements are essentially the original laws of cosine.

Gcos[-3:]

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[a^{2} A^{2} - A^{2} b^{2} - 2 A^{2} b c - A^{2} c^{2} + a^{2} - b^{2} + 2 b c - c^{2}, a^{2} B^{2} - b^{2} B^{2} + 2 a B^{2} c + B^{2} c^{2} + a^{2} - b^{2} - 2 a c + c^{2}, a^{2} C^{2} + 2 a b C^{2} + b^{2} C^{2} - c^{2} C^{2} + a^{2} - 2 a b + b^{2} - c^{2}\right]$

However. This time it is not so easy to find a 'useful' and 'elegant' formula among the basis elements.

We have better luck if we add the requirement $\alpha+\beta+\gamma=\pi$:

sumofangles = A*B + B*C + C*A - 1 Icos = Ideal (coslawa,coslawb,coslawc,sumofangles) Gcos = Icos.groebner_basis('libsingular:std') len (Gcos)

$\newcommand{\Bold}[1]{\mathbf{#1}}22$

Gcos[-4].factor()

$\newcommand{\Bold}[1]{\mathbf{#1}}c \cdot a \cdot (a A - A b - A c + a C + b C - c C)$

This translates to \[\frac{\cot\alpha}{-a+b+c} = \frac{\cot{\gamma}}{a+b-c}\] which is the less known law of cotangents

Laws of trigonometry 2

3371 days ago by Kris.Coolsaet