论文部分内容阅读
一次测验,我们出了这样一道试题:已知方程cos2x+sinx=q有实数解,求实数q的范围。归纳起来,学生大致有这么三种做法: 第一种:∵方程cos2x+sinx=q有实数解,∴q必在函数cos2x+sinx的值域内。而函数cos2x+sinx=1-2sin~2x+sinx=9/8-2(sinx-1/4)~2,当sinx=1/4时,有最大值9/8当sinx=-1时,有最小值-2,故值域为〔-2,9/8〕,∴-2≤q≤9/8 第二种:把已知方程化为关于sinx的二次方程:2sin~2x-sinx+q-1=0 (1)
In one quiz, we made a test such that the known equation cos2x+sinx=q has a real solution and the range of real numbers q. To sum up, students generally have three approaches: First: The ∵ equation cos2x + sinx = q has a real solution, and ∴ q must be in the range of the function cos2x + sinx. The function cos2x+sinx=1-2sin~2x+sinx=9/8-2(sinx-1/4)~2. When sinx=1/4, there is a maximum of 9/8 when sinx=-1. There is a minimum value of -2, so the value range is [-2,9/8], ∴-2≤q≤9/8 Second: Turn the known equation into a quadratic equation about sinx: 2sin~2x-sinx +q-1=0 (1)