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Trigonometric invariants are defined for each Weyl group orbit on the rootlattice. They are real and periodic on the coroot lattice. Their polynomialalgebra is spanned by a basis which is calculated by means of an algorithm. Theinvariants of the basis can be used as coordinates in any cell of the corootspace and lead to an exactly solvable model of Sutherland type. We apply thisconstruction to the $F_4$ caseComment: 13 pages, no figures, latex 2epsilon, corrected versio

THE CONSTRUCTION OF TRIGONOMETRIC INVARIANTS FOR WEYL GROUPS AND THE DERIVATION OF THE CORRESPONDING EXACTLY SOLVABLE SUTHERLAND MODELS