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笔者曾利用课外时间组织学生练习过一道有趣的题目: 设f(sinx)=cos3x(1)求证f(cosx)==-sin3x。此题抄出约10分钟,当堂几乎没有学生能够证出。经观察发现,原来学生们均企图由(1)式先求出f(x)的表达式,再求f(cosx)一一从而陷入了难以摆脱的困境。于是笔者启发学生:能否不求f(x),而将cosx用与它恒等的sib(π/2-x)代替?这时学生的思路由“山穷
The author used the extracurricular time to organize students to practice an interesting topic: Let f(sinx)=cos3x(1) verify that f(cosx)==-sin3x. This question was copied for about 10 minutes, and almost no students in the hall could prove it. It has been observed that the original students all attempted to find the expression of f(x) first by formula (1) and then f(cosx) to get into a difficult position to escape. So the author inspires students: Can we not use f(x) instead of replacing cosx with its equal sib(π/2-x)?