f(x)=cos(2X-π/3)+sin(x-π/4)sin(x+π/4)的最小正周期和图像的对称轴方程是什么?在【-π/12,π/2】的值域是多少?
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/05 13:02:20
![f(x)=cos(2X-π/3)+sin(x-π/4)sin(x+π/4)的最小正周期和图像的对称轴方程是什么?在【-π/12,π/2】的值域是多少?](/uploads/image/z/3913208-8-8.jpg?t=f%28x%29%3Dcos%282X-%CF%80%2F3%29%2Bsin%28x-%CF%80%2F4%29sin%28x%2B%CF%80%2F4%29%E7%9A%84%E6%9C%80%E5%B0%8F%E6%AD%A3%E5%91%A8%E6%9C%9F%E5%92%8C%E5%9B%BE%E5%83%8F%E7%9A%84%E5%AF%B9%E7%A7%B0%E8%BD%B4%E6%96%B9%E7%A8%8B%E6%98%AF%E4%BB%80%E4%B9%88%3F%E5%9C%A8%E3%80%90-%CF%80%2F12%2C%CF%80%2F2%E3%80%91%E7%9A%84%E5%80%BC%E5%9F%9F%E6%98%AF%E5%A4%9A%E5%B0%91%3F)
f(x)=cos(2X-π/3)+sin(x-π/4)sin(x+π/4)的最小正周期和图像的对称轴方程是什么?在【-π/12,π/2】的值域是多少?
f(x)=cos(2X-π/3)+sin(x-π/4)sin(x+π/4)的最小正周期和图像的对称轴方程是什么?在【-π/12,π/2】的值域是多少?
f(x)=cos(2X-π/3)+sin(x-π/4)sin(x+π/4)的最小正周期和图像的对称轴方程是什么?在【-π/12,π/2】的值域是多少?
首先将f(x)化简
f(x)=cos(2X-π/3)+sin(x-π/4)sin(x+π/4)
=cos(2X-π/3)+1/2*(-cos(2x)+cos(-π/2))
=cos(2X-π/3)+1/2*(-cos(2x)+cos(-π/2))
=cos(2X-π/3)-1/2*cos(2x)
=cos(2X)cos(-π/3)-sin(2x)sin(-π/3)-1/2*cos(2x)
=1/2*cos(2X)+sin(2x)sin(π/3)-1/2*cos(2x)
=sin(2x)sin(π/3)
化简后可以看出sin(π/3)是常数,只要求出sin(2x)的值域即可.
可知当x∈【-π/12,π/2】时,2x∈【-π/6,π】
通过正弦函数曲线可知sin2x的最小值是-1/2,最大值是1
故f(x)的最小值是-1/2sin(π/3),最大值是sin(π/3)
f(x)=cos(2x-π/3)+sin(x-π/4)sin(x+π/4)
=cos(2x)/2+sin(2x)sqrt(3)/2+[sin(x)-cos(x)]sqrt(2)/2*[sin(x)+cos(x)]sqrt(2)/2
=cos(2x)/2+sin(2x)sqrt(3)/2+[sin(x)]^2-[cos(x)]^2
=cos(2x...
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f(x)=cos(2x-π/3)+sin(x-π/4)sin(x+π/4)
=cos(2x)/2+sin(2x)sqrt(3)/2+[sin(x)-cos(x)]sqrt(2)/2*[sin(x)+cos(x)]sqrt(2)/2
=cos(2x)/2+sin(2x)sqrt(3)/2+[sin(x)]^2-[cos(x)]^2
=cos(2x)/2+sin(2x)sqrt(3)/2-cos(2x)
=sin(2x)sqrt(3)/2-cos(2x)/2
=sin(2x+2π/3),
故f(x)的最小正周期为2π/2=π,
图像的对称轴方程2x+2π/3=2kπ ==> x=kπ-π/3, k为整数.
在【-π/12,π/2】上,t=2x+2π/3属于【π/2,5π/3】, sin(t)单调减,
故f(x)属于【-sqrt(3)/2,1】。
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