A note on trigonometric identities
If \(g(u,v)\in \mathbb R[u,v]\) is a polynomial such that \(g(\cos x, \sin x) = 0\) for all real numbers \(x\), then it is known that \(g(u,v)\) is a multiple of \(u^2 + v^2 - 1\). It follows that this result is equivalent to the statement that if \(g(u,v)\) is a polynomial such that \(g(\cos,\cos y)\) vanishes whenever \(x,y\) are the acute angles of an arbitrary right-angled triangle, then \(g(u,v)\) is a multiple of \(u^2 + v^2 - 1\). This motivates the authors to ask about what polynomial defines how the cosines of the three angles of a general triangle are related. The authors give two results which provide answers to this and related questions. The first result is as follows: Let \(P = \{(x,y,z)\in \mathbb R^3 : x+y+z =\pi\}\), \(H = u^2 + v^2 +w^2 +2uvw - 1\), and \(K= \{f\in\mathbb R[u,v,w] : f(\cos x,\cos y, \cos z) = 0\) for all \((x,y,z)\in\mathbb P\}\). Then \(K\) is the principal ideal generated by \(H\). The second is given by the following: If \(g\in R[U, V, W]\) is a polynomial such that \(g(\sin x, \sin y,\sin z)\) vanishes for all \((x,y,z)\in P\), then \(g\) is a multiple of \(H_1\), where \(H_1(U,V,W)=(U^4+V^4+W^4)-2(U^2 V^2+V^2W^2+W^2U^2)+ 4U^2V^2W^2\).
scientific article; zbMATH DE number 2065408
scientific article