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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
A
2
=
2
5
5
，
AB
•
AC
=3，b+c=6
（I）求a的值；
（II）求
2sin(A+
π
4
)sin(B+C+
π
4
)
1-cos2A
的值．
答案

（I）∵cos

A
2
=
2
5
5
，

∴cosA=2cos2

A
2
-1=
3
5
，

又

AB
•
AC
=3，即bccosA=3，

∴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+
π
4
)sin(B+C+
π
4
)
1-cos2A

=

2sin(A+
π
4
)sin(π-A+
π
4
)
1-cos2A
=
2sin(A+
π
4
)sin(A-
π
4
)
1-cos2A

=

2sin(A+
π
4
)cos[
π
2
-(A-
π
4
)]
1-cos2A
=
-2sin(A+
π
4
)cos(A+
π
4
)
1-cos2A

=-

sin(2A+
π
2
)
1-cos2A
=-
cos2A
1-cos2A
，

∴cosA=

3
5
，∴cos2A=2cos2A-1=-
7
25
，

∴原式=-

-
7
25
1+
7
25
=
7
32
．

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小题6:March 8 is _____________(woman) Day.

小题7:Thanks for _________(you) video.

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