论文部分内容阅读
1956年第12期《数学通报》上曾发表过阿意今斯他和別罗郭夫斯卡娅写的“求三角函数的周期”一文(由张鉴卿譯自苏联“中学数学”),該文提出了求f_1(x)=cos(3/2)x-sin(x/3),f_2(x)=cos 2x-tgx的周期的問題。本文打算就这些問題加以推广,进而求sin nx+cos mx的周期(其中m,n为实数)。分析:若該函数存在周期b(b>0),則根据周期函数的周期的定义,f(x+b)=sin n(x+b)+cos m(x+b) =sin(nx+nb)+cos(mx+mb) =sin nx+cos mx=f(x). 現在的任务是判断b是否存在;如果存在,如何把它求出来。根据三角函数的性貭知道,对sin nx来說,要使sin(nx+nb)=sin nx对一切x的值都成立,則
In the 12th issue of Mathematical Bulletin of 1956, there was a paper entitled “The Cycle of the Trigonometric Functions” written by Arjizstan and Birogovskaya (translated by Zhang Jianqing from the Soviet Union “Middle School Mathematics”). The problem of finding the cycle of f_1(x)=cos(3/2)x-sin(x/3),f_2(x)=cos 2x-tgx is proposed. This article intends to promote these issues, and then find the cycle of sin nx + cos mx (where m, n is a real number). Analysis: If the function has a period b (b>0), then according to the definition of the period of the periodic function, f(x+b)=sin n(x+b)+cos m(x+b)=sin(nx+ Nb) + cos(mx + mb) = sin nx + cos mx = f(x). Now the task is to determine if b exists; if so, how to find it. According to the property of the trigonometric function, for sin nx, to make sin(nx+nb)=sin nx true for all x values,