高中三角恒等变换.数学帝们快来吧.
来源:学生作业帮 编辑:百度作业网作业帮 分类:数学作业 时间:2024/08/19 14:59:10
高中三角恒等变换.数学帝们快来吧.
![高中三角恒等变换.数学帝们快来吧.](/uploads/image/z/19430381-29-1.jpg?t=%E9%AB%98%E4%B8%AD%E4%B8%89%E8%A7%92%E6%81%92%E7%AD%89%E5%8F%98%E6%8D%A2.%E6%95%B0%E5%AD%A6%E5%B8%9D%E4%BB%AC%E5%BF%AB%E6%9D%A5%E5%90%A7.)
f(x)=sinx+cosx,f'(x)=cosx-sinx.
g(x)=f(x)*f'(x)+[f(x)]^2.
=(cosx+sinx)(cosx-sinx)+(sinx+cosx)^2.
=cos^2x-sin^2x+sin2x+1.
=cos2x+sin2x+1.
∴g(x)=√2sin(2x+π/4)+1.
(1) g(x)的最小正周期T=2π/2=π;
g(x)max=√2+1 【sin(2x+π/4)=1时】.
(2) g(x)的低调递增区间:
∵sinx的单调递增区间为:x∈[2kπ-π/2,2kπ+π/2]
2x∈[4kπ-π,4kπ+π].
2x+π/4∈[4kπ-3π/4,4kπ+5π/4].
∴g(x)的单调递增区间为:(2x+π/4)∈[4kπ-3π/4,4kπ+5π/4].
g(x)=f(x)*f'(x)+[f(x)]^2.
=(cosx+sinx)(cosx-sinx)+(sinx+cosx)^2.
=cos^2x-sin^2x+sin2x+1.
=cos2x+sin2x+1.
∴g(x)=√2sin(2x+π/4)+1.
(1) g(x)的最小正周期T=2π/2=π;
g(x)max=√2+1 【sin(2x+π/4)=1时】.
(2) g(x)的低调递增区间:
∵sinx的单调递增区间为:x∈[2kπ-π/2,2kπ+π/2]
2x∈[4kπ-π,4kπ+π].
2x+π/4∈[4kπ-3π/4,4kπ+5π/4].
∴g(x)的单调递增区间为:(2x+π/4)∈[4kπ-3π/4,4kπ+5π/4].