Question

設(shè)函數(shù)f(x)=a·b,其中向量a=(2-4sin²,1),b=(cosx,3sin2x)(x∈R).

(1)若f(x)=1-,且x∈［,］,求x;

(2)若函數(shù)y=2sin2x的圖象按向量c＝(m,n)(|m|＜)平移后得到函數(shù)y=f(x)的圖象，求實(shí)數(shù)m,n的值.

Answer 1

解:(1)∵f(x)=a·b=(2-4sin²)cosx+sin2x=2cos²x+sin2x=2sin(2x+)+1,

若f(x)=1-,∴2sin(2x+)+1=1-,sin(2x+)=.

又∵≤x≤,∴-≤2x+≤.∴2x+=,即x=.

(2)設(shè)y=f(x)上任意點(diǎn)P(x,y),它是y=2sin2x上的點(diǎn)P₀(x₀,y₀)

按c=(m,n)平移而得.∴

∴y-n=2sin［2(x-m)］,y=2sin(2x-2m)+n.

又y=f(x)=2sin(2x+)+1,∴n=1,-2m=2kπ+,k∈Z.

∴m=-kπ,k∈Z.又|m|＜,∴k=0.

故實(shí)數(shù)m=,n=1.