Discussion Overview
The discussion revolves around the simplification of the trigonometric expression $$\cos[k_n(L-\varepsilon)]-\cos k_nL$$, particularly focusing on the implications of small epsilon in the context of trigonometric identities and Taylor expansions. The scope includes mathematical reasoning and technical explanations related to trigonometric functions.
Discussion Character
- Technical explanation, Mathematical reasoning, Debate/contested
Main Points Raised
- Some participants propose that the expression can be simplified to $$2\sin k_nL$$ using a trigonometric identity for small epsilon.
- Others argue that this simplification is incorrect and suggest it approximates to $$\sim k_n \varepsilon\, \sin(k_n L)$$ instead.
- A participant inquires about the specific trigonometric identity used in the proposed simplification.
- Another participant mentions the cosine of a sum formula and notes that a Taylor expansion of the first term yields the same result.
- There is a suggestion that using the cosine sum formula leads to $$\sin k_nL\sin k_n\varepsilon$$, assuming small epsilon.
- A participant recalls that $$\sin\theta\approx\theta$$ for small theta, which may relate to the approximations being discussed.
Areas of Agreement / Disagreement
Participants do not appear to reach a consensus, as there are competing views on the correct simplification of the expression and the appropriate mathematical approach to take.
Contextual Notes
The discussion includes assumptions about the behavior of trigonometric functions for small angles, and the validity of different mathematical identities and expansions is not resolved.