已知函数f(x)=sin2x+23sinxcosx+3cos2x.(1)求函数f(x)的单调增区间;(2)已知f(α)=3,且α∈(0,π),求α的值.-数学

题目简介

已知函数f(x)=sin2x+23sinxcosx+3cos2x.(1)求函数f(x)的单调增区间;(2)已知f(α)=3,且α∈(0,π),求α的值.-数学

题目详情

已知函数f(x)=sin2x+2
3
sinxcosx+3cos2x

(1)求函数f(x)的单调增区间;
(2)已知f(α)=3,且α∈(0,π),求α的值.
题型:解答题难度:中档来源:台州二模

答案

(1)f(x)=
3
sin2x+cos2x+2
=2sin(2x+class="stub"π
6
)+2

-class="stub"π
2
+2kπ≤2x+class="stub"π
6
≤class="stub"π
2
+2kπ

-class="stub"π
3
+kπ≤x≤class="stub"π
6
+kπ
;.
∴函数f(x)的单调增区间为[-class="stub"π
3
+kπ ,  class="stub"π
6
+kπ ]  (k∈Z)

(2)由f(α)=3,得2sin(2α+class="stub"π
6
)+2=3

sin(2α+class="stub"π
6
)=class="stub"1
2

2α+class="stub"π
6
=class="stub"π
6
+2k1π
,或2α+class="stub"π
6
=class="stub"5π
6
+2k2π
(k1,k2∈Z),
即α=k1π或α=class="stub"π
3
+k2π
(k1,k2∈Z).∵α∈(0,π),
α=class="stub"π
3

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