问题
解答题
在△ABC中,∠A、∠B、∠C所对的边分别为a、b、c,且满足cos
(I)求a的值; (II)求
|
答案
(I)∵cos
=A 2
,2 5 5
∴cosA=2cos2
-1=A 2
,3 5
又
•AB
=3,即bccosA=3,AC
∴bc=5,又b+C=6,
∴b=5,c=1或b=1,c=5,
由余弦定理得:a2=b2+c2-2bccosA=20,
∴a=2
;5
(II)2sin(A+
)sin(B+C+π 4
)π 4 1-cos2A
=
=2sin(A+
)sin(π-A+π 4
)π 4 1-cos2A 2sin(A+
)sin(A-π 4
)π 4 1-cos2A
=
=2sin(A+
)cos[π 4
-(A-π 2
)]π 4 1-cos2A -2sin(A+
)cos(A+π 4
)π 4 1-cos2A
=-
=-sin(2A+
)π 2 1-cos2A
,cos2A 1-cos2A
∴cosA=
,∴cos2A=2cos2A-1=-3 5
,7 25
∴原式=-
=- 7 25 1+ 7 25
.7 32