论文部分内容阅读
在三角教学过程中,有些同志提出,有些三角函数周期是观察得出的,总感到说服力不强。如函数y=|sinx|+|cosx|;y=ctgx-tgx;y=sin(3/5)x+cos2x;y=sin~2x;……它们的周期怎么求,又如,怎样判断函数y=x+sinx,y=sin~(1/2)x,y=sinx~2……是不是周期函数。这里,我提供一个方法;供大家研究参考。这个方法的基本思想是透彻的理解和应用周期函数的定义。为了求出函数y=f(x)的周期,或判定它是否为周期函数,我们讨论对于常数T>0,等式f(x+T)=f(x)对于函数y=f(x)的定义域中的一切实数x是否成立。由f(x+T)-f(x)=0,左边化成若干个因子的乘积:f_1(x,T),f_2(x,T)…f_n(x,T)=0若要f(x+T)-f(x)=0对定义域中一切x都成立,则至少有一因子f_1(x,T)=0对于一切x恒成立。由此分析,对一般的三角函数,可判定它是否为周期函数,若y=f(x)为周期函数,也可以由此求出其周期。现举例说明。
In the process of teaching the trigonometry, some comrades proposed that some trigonometric function cycles are observed and always feel less convincing. If the function y=|sinx|+|cosx|;y=ctgx-tgx;y=sin(3/5)x+cos2x;y=sin~2x;......How do their periods behave, and how do we determine the function y=x+sinx,y=sin~(1/2)x,y=sinx~2... Is it a periodic function? Here, I provide a method; for everyone’s reference. The basic idea of this method is to thoroughly understand and apply the definition of the periodic function. To find the period of the function y=f(x), or to determine whether it is a periodic function, we discuss that for a constant T>0, the equation f(x+T)=f(x) for the function y=f(x) Whether all real numbers x in the definition field of the definition are true. From f(x+T)-f(x)=0, the left side is multiplied by the product of several factors: f_1(x,T),f_2(x,T)...f_n(x,T)=0 if f(x) +T)-f(x)=0 holds for all x in the domain, and at least one factor f_1(x,T)=0 holds true for all x constants. From this analysis, for a general trigonometric function, it can be determined whether it is a periodic function, and if y=f(x) is a periodic function, its period can also be found therefrom. Here is an example.