The combinatorial structure of trigonometry
The native mathematical language of trigonometry is combinatorial. Two interrelated combinatorial symmetric functions underlie trigonometry. We use their characteristics to derive identities for the trigonometric functions of multiple distinct angles. When applied to the sum of an infinite number of...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203106230 |
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| Summary: | The native mathematical language of trigonometry is
combinatorial. Two interrelated combinatorial symmetric functions
underlie trigonometry. We use their characteristics to derive
identities for the trigonometric functions of multiple distinct
angles. When applied to the sum of an infinite number of
infinitesimal angles, these identities lead to the power series
expansions of the trigonometric functions. When applied to the
interior angles of a polygon, they lead to two general constraints
satisfied by the corresponding tangents. In the case of multiple
equal angles, they reduce to the Bernoulli identities. For the
case of two distinct angles, they reduce to the Ptolemy identity.
They can also be used to derive the De Moivre-Cotes identity. The
above results combined provide an appropriate mathematical
combinatorial language and formalism for trigonometry and more
generally polygonometry. This latter is the structural language
of molecular organization, and is omnipresent in the natural
phenomena of molecular physics, chemistry, and biology.
Polygonometry is as important in the study of moderately complex
structures, as trigonometry has historically been in the study of
simple structures. |
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| ISSN: | 0161-1712 1687-0425 |