International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 4, Pages 705-714

Trigonometric identities

Malvina Baica

Department of Mathematics and Computer Science, University of Wisconsin, Whitewater 53190, Wisconsin, USA

Received 20 November 1985

Copyright © 1986 Malvina Baica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper the author obtains new trigonometric identities of the form 2(p1)(p2)2k=1p2(1cos2πkp)p1k=pp2 which are derived as a result of relations in a cyclotomic field (ρ), where is the field of rationals and ρ is a root of unity.

Those identities hold for every positive integer p3 and any proof avoiding cyclotomic fields could be very difficult, if not insoluble. Two formulask=1p12(1)(p2k)tanp12kϕ=0and1+k=0p12(1)k(i=0p12k2(p2k+2i)(k+1k))cosp2kϕ=0stated only by Gauss in a slightly different form without a proof, are obtained and used in this paper in order to give some numeric applications of our new trigonometric identities.