急>< TAT已知向量a=(sinx,1),向量b=(cosx,-1/2) (1)当a⊥b,求丨a+b丨的值 (2)求函数f(x)=a·(a-b)的值域
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急>< TAT已知向量a=(sinx,1),向量b=(cosx,-1/2) (1)当a⊥b,求丨a+b丨的值 (2)求函数f(x)=a·(a-b)的值域
急>< TAT
已知向量a=(sinx,1),向量b=(cosx,-1/2) (1)当a⊥b,求丨a+b丨的值 (2)求函数f(x)=a·(a-b)的值域
急>< TAT已知向量a=(sinx,1),向量b=(cosx,-1/2) (1)当a⊥b,求丨a+b丨的值 (2)求函数f(x)=a·(a-b)的值域
a⊥b时,a,b构成直角三角形,
|a+b|^2=|a|^2+|b|^2=(sinx^2+1)+(cosx^2+1/4)=9/4
=>|a+b|=3/2
f(x)
=a(a-b)
=a^2-a.b
=sinx^2+1-(sinxcosx-1/2)
=(1-cos2x)/2-sin2x/2+3/2
=2-(sin2x+cos2x)/2
=2-√2/2sin(2x+π/4)
∈[2-√2/2,2+√2/2]
(1)a⊥b 时,x=π/4,所以|a+b|=π²/4-1/2
(2)值域为[1-√2/2,1+√2/2]
(1)a⊥b,a*b=0 sinxcosx-1/2=0 sinxcosx=1/2 (a+b)*(a+b)=(sinx+cosx)^2+1/4=sin^2x+2sinxcosx+cos^2x+1/4=1+2/2+1/4=2+1/4=9/4
丨a+b丨=根号(a+b)*(a+b)=根号9/4 =3/2
(2)a-b=(sinx-cosx,3/2)
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(1)a⊥b,a*b=0 sinxcosx-1/2=0 sinxcosx=1/2 (a+b)*(a+b)=(sinx+cosx)^2+1/4=sin^2x+2sinxcosx+cos^2x+1/4=1+2/2+1/4=2+1/4=9/4
丨a+b丨=根号(a+b)*(a+b)=根号9/4 =3/2
(2)a-b=(sinx-cosx,3/2)
a·(a-b)=sin^2x-sinxcosx+3/2=1/2-cos2x/2-sin2x/2+3/2=2-(cos2x+sin2x)/2=2-根号2(根号2cos2x/2+根号2sin2x/2)/2=2-sin(2x+π/4)/4 -1/4 <=-sin(2x+π/4)/4 <=1/4
7/4 <=2-sin(2x+π/4)/4 <=9/4 函数f(x)=a·(a-b)的值域 7/4<=f(x)<=9/4
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