(1)已知tanα=23,1sin2α-2sinαcosα+4cos2α的值.(2)已知π4<α<3π4,0<β<π4,且cos(π4-α)=35,sin(π4+β)=513,求sin(α+β)的值.

题目简介

(1)已知tanα=23,1sin2α-2sinαcosα+4cos2α的值.(2)已知π4<α<3π4,0<β<π4,且cos(π4-α)=35,sin(π4+β)=513,求sin(α+β)的值.

题目详情

(1)已知tanα=
2
3
1
sin2α-2sinαcosα+4cos2α
的值.
(2)已知
π
4
<α<
4
,0<β<
π
4
,且cos(
π
4
-α)=
3
5
,sin(
π
4
+β)=
5
13
,求sin(α+β)的值.
题型:解答题难度:中档来源:不详

答案

(1)class="stub"1
sin2α-2sinαcosα+4cos2α
=
sin2α+cos2α
sin2α-2sinαcosα+4cos2α
=
tan2α+1
tan2α-2tanα+4

tanα=class="stub"2
3
,∴
tan2α+1
tan2α-2tanα+4
=class="stub"13
28

(2)∵class="stub"π
4
<α<class="stub"3π
4
,0<β<class="stub"π
4
,且cos(class="stub"π
4
-α)=class="stub"3
5
,sin(class="stub"π
4
+β)=class="stub"5
13

∴sin(class="stub"π
4
-α)=-class="stub"4
5
,cos(class="stub"π
4
+β)=class="stub"12
13

∴sin(α+β)=sin[(class="stub"π
4
+β)-(class="stub"π
4
-α)]=class="stub"5
13
•class="stub"3
5
-class="stub"12
13
•(-class="stub"4
5
)
=class="stub"63
65

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