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Extended Sine and Cosine Functions for Scalene Triangles

This article pushes the role of sine and cosine functions beyond the traditional purpose of determining the sides of a right triangle, into the realm of determining the lengths of the sides of any triangle with practically the same ease. Extended functions are formulated dependent on two angles (instead of the traditional one) — sin*(α,γ) and cos*(α,γ)—that allow (via direct application) the computation of the lengths of the two shorter sides of a scalene triangle, as a result of the angular projection (from reference angle γ and a variable obtuse angle ) of the longer side or extended hypotenuse (for right triangles, the obtuse angle is fixed to γ= 90 deg, allowing only the variation of α— a significant limitation). When integrated into larger more complex mathematical formula, the extended sine and cosine functions add greater flexibility and open the door for the mathematician or scientist to explore possibilities that are non-orthogonal. Solved exercises are provided at the end, with the purpose of illustrating the robustness and advantage of the application of these new extended sine and cosine functions to determine the normalized sides of a scalene triangle—a requirement that is present virtually in any technical discipline.