Extended Angle Sum and Difference Identity Rules for Scalene Triangles

  •  Luis Teia    


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.

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