Assistance for the Calculation of Sines (Translation of E246)

Paralleling his famous relation eiφ = cos(φ) + i sin(φ), Euler establishes the equality (cos φ + i sin φ)n = (cos nφ + i sin nφ). He uses it to comprehensively derive trigonometric identities that convert arbitrary powers of sines and cosines of an angle (and products thereof) into sums of sines and cosines of multiples of that angle. Some negative and fractional powers are shown to yield infinite series. Euler further describes a general method to evaluate various infinite series involving weighted trigonometric functions. These results foreshadow Fourier series. As Euler points out, the scope of applicability of the proposed techniques extends far beyond the specific results given herein.