若A+B=2π3,则cos2A+cos2B的取值范围是______.-数学

题目简介

若A+B=2π3,则cos2A+cos2B的取值范围是______.-数学

题目详情

A+B=
3
,则cos2A+cos2B
的取值范围是______.
题型:填空题难度:中档来源:不详

答案

cos2A+cos2B
=class="stub"1
2
(2cos2A-1)+class="stub"1
2
+class="stub"1
2
(2cos2B-1)+class="stub"1
2

=class="stub"1
2
cos2A+class="stub"1
2
cos2B+1
A+B=class="stub"2π
3

∴B=class="stub"2π
3
-A
class="stub"1
2
cos2A+class="stub"1
2
cos2B+1
=class="stub"1
2
cos2A+class="stub"1
2
cos(class="stub"4π
3
-2A)+1
=class="stub"1
2
cos2A+class="stub"1
2
[(-class="stub"1
2
cos2A)-
3
2
sin2A]+1
=class="stub"1
2
class="stub"1
2
cos2A-
3
2
sin2A)+1
=class="stub"1
2
cos(2A+class="stub"π
3
)+1
即cos2A+cos2B=class="stub"1
2
cos(2A+class="stub"π
3
)+1
∵-1≤cos(2A+class="stub"π
3
)≤1
class="stub"1
2
class="stub"1
2
cos(2A+class="stub"π
3
)+1≤class="stub"3
2

即cos2A+cos2B的取值范围为[class="stub"1
2
,class="stub"3
2
]

故答案为:[class="stub"1
2
,class="stub"3
2
]

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