On Trigonometric Formulations of Polynomial Equations

The displacement analysis of open and closed kinematic chains is based on polynomial equations whose variables are functions of relative joint displacements. The objective of this paper is to investigate new and interesting properties of the transformations between the canonical cosine-sine polynomials and the even degree tan-half angle polynomials associated with displacement kinematics. Using a homogeneous coordinate formulation, it is shown that the coefficients of the polynomials are linearly related by a projective transformation whose elements can be defined recursively. The canonical cosine-sine polynomial is then transformed to a cosine or a sine polynomial which can be rooted by usual techniques. However, all real roots are bracketed between −1 and +1 which can have numerical advantages over a corresponding tan-half angle polynomial for which the entire real axis must be searched. It is also demonstrated how polynomial solutions corresponding to circular points at infinity in the tan-half angle, which are typically introduced as extraneous roots via algebraic elimination, may be easily factored out by the transformation to the cosine-sine formulation.

A SHORT NOTE ON TWO INEQUALITIES FOR SINE POLYNOMIALS

We present elementary proofs for \[\sum_{\nu=1}^n(n+1-\nu)\sin(\nu x)>0\] due to Lukács, and for \[\sum_{\nu=1}^n\sin(\nu x)+\frac{1}{2}\sin((n+1)x) \quad\quad (*) \] due to Fejér. Both inequalities are valid for $x \in (0, \pi )$ and $n = 1, 2, \cdots$. Furthermore we determine all cases of equality in (*).