[ARTICLE] Extended Angle Sum and Difference Identity Rules for Scalene Triangles (Canadian Journal of Mathematics Research / 2022) - GAMMA TRIGONOMETRY
Bibliographic record
Abstract
The present article extends the applicability of the angle sum and difference identity rules sin(A±B) and cos(A±B) beyond the particular case of a right-angled triangle into the general case of scalene triangles as sin*(A±B,γ) and cos*(A±B,γ), adding the effect of independently varying both the reference angle α = A±B and the obtuse angle γ. Accompanied by appropriate theorems and proofs, the mathematical end result are four updated equations that supersede the traditional expressions sin(A±B) = sin(A)cos(b)± cos(A)sin(B) and cos(A±B) = cos(α)cos(B) ∓ sin(A) sin(B), where the conventional sin(α) and cos(α) functions are replaced by the [already proven] extended versions sin*(α,γ) and cos*(α,γ) enclosing modifications including two angles α and γ. An open-source program scripted in Octave is provided for the verification of the derived expressions, including plotting the geometric results as a figure for both the cases of angle summation and subtraction.NOTE: The second link at the bottom provides access to the article about the extended sine function sin*(α,γ) and extended cosine function cos*(α,γ). 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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.028 | 0.013 |
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".