Subsidium calculi sinuum

English Title

A contribution to the calculations of sines

Authors

Leonhard Euler

Enestrom Number

246

Fuss Index

329

Original Language

Latin

Archive Notes

According to Jacobi, a treatise with the title "Subsidium doctrinae sinuum" was read to the Berlin Academy on March 9, 1752. According to St. Petersburg Academy records, it was presented to that Academy on March 12, 1753.

Content Summary

Let u = cos(φ†) + i sin(φ) and v = cos(φ†) – i sin(φ†). Euler shows these are e^iφ† and e^-iφ†, respectively. He uses these to find cos^m(φ†), sin^m(φ†), and their product cos^m(φ†)∙sin^m(φ†). He sums series of complex numbers to get more things that look like Fourier series, including the series φ†/2 = sin(φ†) – sin(2φ†)/2 + sin(3φ†)/3 – sin(4φ†)/4 ± .... Apparently, we may also find this theorem in E447 and E655, as well as in works by Cauchy and Abel.

Published as

Journal article

Published Date

1760

Written Date

1752

Original Source Citation

Novi Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 164-204.

Opera Omnia Citation

Series 1, Volume 14, pp.542-584.

Record Created

2018-09-25