苏教版高中数学教材必须修读4第1课时三角函数.ppt
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1.3.1三角函数的周期性
金陵中学 金凤义
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【教学目标】
(1)了解周期现象在现实中广泛存在,感受周期现象对实际工作的意义;
(2)了解周期函数的概念,会判断一些简单的、常见的函数的周期性,并会求一些简单三角函数的周期;
(3)培养及渗透数形结合思想,培养辩证唯物主义观点.
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(一)情境引入
1.问题:
(1)今天是星期二,则过了七天是星期几?过了十四天呢?……
(2)物理学中的单摆振动、圆周运动中质点运动,规律如何呢?
2.我们学过的函数中哪些函数也具有这种“周而复始”的基本特征呢?怎样从数学的角度研究函数的周期现象呢?
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(二)意义建构
由单位圆中的三角函数线可知,正、余弦函数值的变化呈现出周期现象,每当角增加(或减少)2π,所得角的终边与原来角的终边相同,故两角的正、余弦函数值也分别相同.即有sin(2π+x)=sinx,cos(2π+x)=cosx,
正弦函数和余弦函数所具有的这种性质称为周期性.
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(三)数学理论
一般地,对于函数f(x),如果存在一个非零常数T,使得定义域内的每一个x值,都满足f(x+T)=f(x),那么函数f(x)就叫做周期函数,非零常数T叫做这个函数的周期.
对于一个周期函数f(x),如果在它所有的周期中存在一个最小的正数,那么这个最小正数就叫做f(x)的最小正周期.
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(四)数学应用
例1
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