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求型如 y=a_1sinx+b_1cosx+c_1/a_2sinx+b_2cosx+c_2的函数值域,常规解法一般有两种,一是把原函数变形为 sin(x+(?))=F(y)型,然后利用三角函数的有界性解不等式|F(y)|≤1(通常为无理不等式);二是利用万能公式变形转化为关于 tan(x/2)的二次方程,利用二次方程的判别式求解.这两种解法固然可行,但过程繁琐、冗长.下面介绍一种新的方法——三角方程“判别式”法,首先我们证明一个定理.
For example, if the type of function is y=a_1sinx+b_1cosx+c_1/a_2sinx+b_2cosx+c_2, there are two general solutions. One is to transform the original function to sin(x+(?))=F(y), then Boundedness solution inequality using trigonometric functions |F(y)| ≤1 (usually irrational inequality); second is the use of universal formula deformation to convert to quadratic equations about tan(x/2), using quadratic equations to discriminate Solving the equation. These two solutions are of course feasible, but the process is cumbersome and tedious. The following describes a new method—trigonometric “discriminant” method. First we prove a theorem.