已知向量a=(2sin(x+π12),cos(x-π12),b=(cos(x+π12),2sin(x-π12)),函数f
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已知向量
a |
![已知向量a=(2sin(x+π12),cos(x-π12),b=(cos(x+π12),2sin(x-π12)),函数f](/uploads/image/z/18068678-62-8.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%EF%BC%882sin%EF%BC%88x%2B%CF%8012%EF%BC%89%EF%BC%8Ccos%EF%BC%88x-%CF%8012%EF%BC%89%EF%BC%8Cb%3D%EF%BC%88cos%EF%BC%88x%2B%CF%8012%EF%BC%89%EF%BC%8C2sin%EF%BC%88x-%CF%8012%EF%BC%89%EF%BC%89%EF%BC%8C%E5%87%BD%E6%95%B0f)
(1)∵f(x)=
a•
b-2cos2x
=2sin(x+
π
12)•cos(x+
π
12)+2sin(x-
π
12)•cos(x-
π
12)-2cos2x
=sin(2x+
π
6)+sin(2x-
π
6)-2cos2x
=
3sin2x-cos2x-1
=2sin(2x-
π
6)-1,
∴f(x)=2sin(2x-
π
6)-1的最小正周期T=
2π
2=π;
(2)g(x)=f(x+
π
4)-1
=
3sin[2(x+
π
4)-
π
6]-1-1
=
3sin(2x+
π
3)-2,
∵x∈[0,
π
2],
∴2x+
π
3∈[
π
3,
4π
3],
∴-
3
2≤sin(2x+
π
3)≤1,
∴g(x)∈[-
7
2,
3-2],
∴g(x)max=
3-2,g(x)min=-
7
2.
a•
b-2cos2x
=2sin(x+
π
12)•cos(x+
π
12)+2sin(x-
π
12)•cos(x-
π
12)-2cos2x
=sin(2x+
π
6)+sin(2x-
π
6)-2cos2x
=
3sin2x-cos2x-1
=2sin(2x-
π
6)-1,
∴f(x)=2sin(2x-
π
6)-1的最小正周期T=
2π
2=π;
(2)g(x)=f(x+
π
4)-1
=
3sin[2(x+
π
4)-
π
6]-1-1
=
3sin(2x+
π
3)-2,
∵x∈[0,
π
2],
∴2x+
π
3∈[
π
3,
4π
3],
∴-
3
2≤sin(2x+
π
3)≤1,
∴g(x)∈[-
7
2,
3-2],
∴g(x)max=
3-2,g(x)min=-
7
2.
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