[ARTICLE] Extended Sine and Cosine Functions for Scalene Triangles (Canadian Journal of Mathematics Research / 2022) - GAMMA TRIGONOMETRY
Bibliographic record
Abstract
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 α and a variable obtuse angle γ) 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. CopyrightsCopyright for this article is retained by the author(s), with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).-------------------------------------------------------------------------For more public data, please visit my Figshare profile: https://figshare.com/authors/Luis_Teia/10811244
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.039 | 0.016 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".