已知函数f(x)=2sin²(π/4+x)-√3cos2x-1,x∈R.若函数h(x)=f(x+t)的图像关于点(-π/6,0)对称,且t属于(0,π/2),则t的值为?
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![已知函数f(x)=2sin²(π/4+x)-√3cos2x-1,x∈R.若函数h(x)=f(x+t)的图像关于点(-π/6,0)对称,且t属于(0,π/2),则t的值为?](/uploads/image/z/5538342-30-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D2sin%26sup2%3B%28%CF%80%2F4%2Bx%29-%E2%88%9A3cos2x-1%2Cx%E2%88%88R.%E8%8B%A5%E5%87%BD%E6%95%B0h%28x%29%3Df%28x%2Bt%29%E7%9A%84%E5%9B%BE%E5%83%8F%E5%85%B3%E4%BA%8E%E7%82%B9%28-%CF%80%2F6%2C0%29%E5%AF%B9%E7%A7%B0%2C%E4%B8%94t%E5%B1%9E%E4%BA%8E%280%2C%CF%80%2F2%29%2C%E5%88%99t%E7%9A%84%E5%80%BC%E4%B8%BA%3F)
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已知函数f(x)=2sin²(π/4+x)-√3cos2x-1,x∈R.若函数h(x)=f(x+t)的图像关于点(-π/6,0)对称,且t属于(0,π/2),则t的值为?
已知函数f(x)=2sin²(π/4+x)-√3cos2x-1,x∈R.若函数h(x)=f(x+t)的图像关于点(-π/6,0)对称,且t属于(0,π/2),则t的值为?
已知函数f(x)=2sin²(π/4+x)-√3cos2x-1,x∈R.若函数h(x)=f(x+t)的图像关于点(-π/6,0)对称,且t属于(0,π/2),则t的值为?
f(x)=2[sin(π/4+x)]^2-√3cos2x-1
=-cos(2x+π/2)-√3cos2x
=sin2x-√3cos2x
=2sin(2x-π/3) x∈R
所以f(x)=2sin(2x-π/3) x∈R
h(x)=f(x+t)的图象关于点(-π/6,0)对称
那么h(x-t)=f(x)的图象关于点(-π/6 + t,0)对称
而f(x)=2sin(2x-π/3)关于点(π/6,0)对称
所以-π/6+t=π/6+2kπ t∈(0,π/2)
t=π/3
f(x)
= 1 - cos(2x + π / 2) - √3cos2x - 1
= sin2x - √3cos2x
= 2sin(2x - π / 3)
f(x + t) = 2sin(2x + 2t - π / 3)
因为图像关于点(-π/6,0)对称
2t - 2π / 3 = nπ, n是整数
t = π / 3