On The Finite Summations Of Trigonometric Functions

In this paper,we study the finite summations of trigonometric functions which including the kπ/n(n∈N+,k=0,0,…,n-1)rad.In this paper,some previous results and methods are collated,improved and generalized,we deduce eight explicit formulae for the known finite summations of high power trigonometric functions again.This paper is mainly divided into two parts namely the sums including angular parameter and excluding angular parameter.In each part we use the generating functions and calculational skills of the combinatorial identities to study the finite summations of cosecant function and cotangent function,and according to the alternation of the sums we divide the sums into the first class finite sum and the second class finite sum.In this paper,we prove the rationality of every finite summation of the trigonometric function,and use differential and limit to increase the power of the trigonometric functions from 1 or 2,in this way,we establish the recursive algorithms.There are three innovative points in this article.1.The parity of the parameter n in some previous results are integrated so that the final explicit formulae are applicable to all positive integers n,which are unified into the first class sum.2.The methods of some previous studies are improved and simplified.For example,in this paper,all the calculations of generating functions are attributed to two trigonometric function identities,we design complex integrals and use the Cauchy integral theorem and the Residue theorem to prove the two trigonometric identities;and in this paper,the parameters of the previous combinatorial identities are extended from integers to real numbers,and integrated into two generalized combinatorial identities,so that the calculations of many binomial coefficients are uniformly simplified.3.A finite summation of high power trigonometric function with two parameters is talked.