问题 填空题
已知sin(α+
π
3
)=
1
3
,且满足α∈[-
π
3
π
6
],则cos2α的值是______.
答案

∵α∈[-

π
3
π
6
],

∴α+

π
3
∈[0,
π
2
],

∵sin(α+

π
3
)=
1
3

∴cos(α+

π
3
)=
1-(
1
3
)2
=
2
2
3

∴cosα=cos[(α+

π
3
)-
π
3
]=cos(α+
π
3
)cos
π
3
+sin(α+
π
3
)sin
π
3
=
2
2
3
×
1
2
+
1
3
×
3
2
=
2
2
+
3
6

则cos2α=2cos2α-1=2×(

2
2
+
3
6
2-1=
4
6
-7
18

故答案为:

4
6
-7
18

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