A contribution to the calculations of sines
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.
Let u = cos(Ï) + i sin(Ï) and v = cos(Ï) – i sin(Ï). Euler shows these are eiÏ and e-iÏ, respectively. He uses these to find cosm(Ï), sinm(Ï), and their product cosm(Ï)∙sinm(Ï). He sums series of complex numbers to get more things that look like Fourier series, including the series Ï/2 = sin(Ï) – sin(2Ï)/2 + sin(3Ï)/3 – sin(4Ï)/4 Â± .... Apparently, we may also find this theorem in E447 and E655, as well as in works by Cauchy and Abel.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 5, pp. 164-204.
Opera Omnia Citation
Series 1, Volume 14, pp.542-584.