已知向量m=(2sinx2,1),n=(cosx2,1),设函数f(x)=m•n-1.(1)求函数y=f(x)的值域;(2)已知△ABC为锐角三角形,A为△ABC的内角,若f(A)=35,求f(2A-

题目简介

已知向量m=(2sinx2,1),n=(cosx2,1),设函数f(x)=m•n-1.(1)求函数y=f(x)的值域;(2)已知△ABC为锐角三角形,A为△ABC的内角,若f(A)=35,求f(2A-

题目详情

已知向量
m
=(2sin
x
2
,1),
n
=(cos
x
2
,1),设函数f(x)=
m
n
-1.
(1)求函数y=f(x)的值域;
(2)已知△ABC为锐角三角形,A为△ABC的内角,若f(A)=
3
5
,求f(2A-
π
3
)的值.
题型:解答题难度:中档来源:不详

答案

(1)由f(x)=
m
n
-1,得f(x)=2sinclass="stub"x
2
cosclass="stub"x
2
+1-1=sinx,
所以y=f(x)的值域为[-1,1];
(2)由已知得A为锐角,f(A)=sinA=class="stub"3
5

则cosA=
1-(class="stub"3
5
)2
=class="stub"4
5
,得sin2A=2sinAcosA=2×class="stub"3
5
×class="stub"4
5
=class="stub"24
25

cos2A=1-2sin2A=1-2×(class="stub"3
5
)2
=class="stub"7
25

所以f(2A-class="stub"π
3
)=sin(2A-class="stub"π
3
)=sin2Acosclass="stub"π
3
-cos2Asinclass="stub"π
3
=class="stub"24
25
×class="stub"1
2
-class="stub"7
25
×
3
2
=
24-7
3
50

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