作业帮(1)若0≤x≤π/2,求函数f(x)=sinx+cosx的值域 (2)若π/3≤x≤2π/3,求函数f(x)=sin^2x+2cosx的值域[1,√2](2)[-1/4,7/4]
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/28 15:47:50
[-1/4,7/4]](/uploads/image/z/2622736-64-6.jpg?t=%281%29%E8%8B%A50%E2%89%A4x%E2%89%A4%CF%80%2F2%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%3Dsinx%2Bcosx%E7%9A%84%E5%80%BC%E5%9F%9F+%282%29%E8%8B%A5%CF%80%2F3%E2%89%A4x%E2%89%A42%CF%80%2F3%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28x%29%3Dsin%5E2x%2B2cosx%E7%9A%84%E5%80%BC%E5%9F%9F%5B1%2C%E2%88%9A2%5D%282%29%5B-1%2F4%2C7%2F4%5D)
(1)若0≤x≤π/2,求函数f(x)=sinx+cosx的值域 (2)若π/3≤x≤2π/3,求函数f(x)=sin^2x+2cosx的值域[1,√2](2)[-1/4,7/4]
(1)若0≤x≤π/2,求函数f(x)=sinx+cosx的值域 (2)若π/3≤x≤2π/3,求函数f(x)=sin^2x+2cosx的值域
[1,√2]
(2)[-1/4,7/4]
(1)若0≤x≤π/2,求函数f(x)=sinx+cosx的值域 (2)若π/3≤x≤2π/3,求函数f(x)=sin^2x+2cosx的值域[1,√2](2)[-1/4,7/4]
【公式】：
① sina * cosb + cosa * sinb = sin(a+b)
② sin^2(a) + cos^2(a) = 1
【解】：
【1】f(x)=sinx + cosx
=√2 * [sinx*cos(π/4) + cosx*sin(π/4)]
=√2 * sin(x+π/4)
因为,0≤x≤π/2
☞ π/4≤x+π/4≤3π/4
☞ √2/2≤sin(x+π/4)≤1
☞1≤√2sin(x+π/4)≤√2
所以,f(x)的值域是 【1,√2】
【2】f(x)=sin^2(x) + 2cosx
=1-cos^2(x) + 2cosx
=-(cosx-1)^2 +2
因为,π/3≤x≤2π/3
☞ -1/2≤cosx≤1/2
☞-3/2≤cos-1≤-1/2
☞ -9/4≤-(cosx-1)^2≤-1/4
可得,-9/4≤-(cosx-1)^2+2≤7/4
所以,f(x)的值域是 【-1/4,7/4】
看图，你也可以化简之后把函数放在坐标上看看图形就知道增减区间，最大最小值
f(x)=sinx+cosx=sqrt(2)*sin(x+45)
所以 若0≤x≤π/2 f(x)=sqrt(2)*sin(45) ~sqrt(2)*sin(90+45)
为一个峰值左右45 故 答案(1) [1,√2]
(2)[-1/4,7/4]
f(x)=sin^2x+2cosx=1-cos^2x+2cosx=-(cosx-1)^2+2
若π/3≤x≤2π/3, cosx=cos60 ~ -cos60=-1/2~ +1/2
答案 (2)[-1/4,7/4]