若α是第三象限角,且sin(α+β)cosβ-sinβcos(α+β)=-1213,则tanα2=______.-数学

题目简介

若α是第三象限角,且sin(α+β)cosβ-sinβcos(α+β)=-1213,则tanα2=______.-数学

题目详情

若α是第三象限角,且sin(α+β)cosβ-sinβcos(α+β)=-
12
13
,则tan
α
2
=______.
题型:填空题难度:中档来源:不详

答案

sin(α+β)cosβ-sinβcos(α+β)=-class="stub"12
13
,得sin[(α+β)-β]=sinα=-class="stub"12
13

则sinα=2sinclass="stub"α
2
cosclass="stub"α
2
=
2sinclass="stub"α
2
cosclass="stub"α
2
sin2class="stub"α
2
+cos2class="stub"α
2
=
2tanclass="stub"α
2
tan2class="stub"α
2
+1
=-class="stub"12
13
,解得tanclass="stub"α
2
=-class="stub"3
2
或-class="stub"2
3

由α是第三象限角,所以2kπ-π<α<2kπ-class="stub"π
2
,k∈Z

kπ-class="stub"π
2
<class="stub"α
2
<kπ-class="stub"π
4
,k∈Z

所以tanclass="stub"α
2
=-class="stub"3
2

故答案为:-class="stub"3
2

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