Hosoya, Haruo Symmetrical combination of the sums and products of trigonometric functions giving integers. (English) Zbl 0880.11024 Symmetry Cult. Sci. 1, No. 4, 431-439 (1990). Systematic methods are given for evaluating various trigonometric sums that have integer values. Some striking examples are \[ \sum^{100}_{k=1} \left(2 \cos {k\pi \over 202} \right)^6 =\sum^{100}_{k=1} \left(2\sin {k\pi\over 202} \right)^6=1988, \]\[ \sum^5_{k= 1} \left(2 \cos {k\pi\over 12} \right)^{10} =\sum^5_{k=1} \left(2 \sin {k\pi \over 12} \right)^{10} = 1000. \] Reviewer: T.M.Apostol (Pasadena) MSC: 11B99 Sequences and sets 33B10 Exponential and trigonometric functions 11L03 Trigonometric and exponential sums (general theory) Keywords:trigonometric sums; integer values PDFBibTeX XMLCite \textit{H. Hosoya}, Symmetry Cult. Sci. 1, No. 4, 431--439 (1990; Zbl 0880.11024)