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.