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-1≤sinx≤1、-1≤cosx≤1是三角函数的重要性质,在解决数学问题中经常发挥很好的作用。但在有些问题中,题设给定或隐含着x的变化范围,使得sinx(或cosx)不能取遍区间[-1,1]内的所有值。就是说,在该问题中,sinx(或cosx)的实际取值范围仅是区间[-1,1]的一个真子集,如果不注意挖掘和运用变量x的范围来确定sinx(或cosx)的实际取值范围用于解决该问题,而盲目套用-1≤sinx≤1、-1≤cosx≤1就会犯错误。因此,应本着具体情况具体分析的精神,加强挖掘和运用题设中所给范围的意识。下面举例说明这个问题。
-1 ≤ sinx ≤ 1, - 1 ≤ cosx ≤ 1 is an important property of the trigonometric function, often play a very good role in solving mathematical problems. However, in some problems, the problem sets a range of variation given or implied by x, so that sinx (or cosx) cannot take all values in the interval [-1,1]. That is, in this problem, the actual range of sinx (or cosx) is only a true subset of interval [-1,1], if you do not pay attention to mining and use the range of variable x to determine the sinx (or cosx) The actual value range is used to solve this problem, and blindly applying -1 ≤ sinx ≤ 1, -1 ≤ cosx ≤ 1 will make mistakes. Therefore, in the spirit of concrete analysis of specific situations, we should strengthen the awareness of excavating and applying the scope given in the title design. The following example illustrates this problem.