已知f(x)=sin2x-cos2x+11+ctgx①化简f(x);②若sin(x+π4)=35,且π4<x<34π,求f(x)的值.-数学

题目简介

已知f(x)=sin2x-cos2x+11+ctgx①化简f(x);②若sin(x+π4)=35,且π4<x<34π,求f(x)的值.-数学

题目详情

已知f(x)=
sin2x-cos2x+1
1+ctgx

①化简f(x);
②若sin(x+
π
4
)=
3
5
,且
π
4
<x<
3
4
π
,求f(x)的值.
题型:解答题难度:中档来源:不详

答案

解.:①f(x)=class="stub"sin2x-cos2x+1
1+ctgx
=
2sinx•cosx-1+2sin2x+1
1+class="stub"cosx
sinx
=
2sin2x•(cosx+sinx)
sinx+cosx
=2sin2x

②∵class="stub"π
4
<x<class="stub"3
4
π
,∴class="stub"π
2
<x+class="stub"π
4
<π
,∴cos(x+class="stub"π
4
)=-
1-sin2(x+class="stub"π
4
)
=-class="stub"4
5

sinx=sin[(x+class="stub"π
4
)-class="stub"π
4
]
=sin(x+class="stub"π
4
)cosclass="stub"π
4
-cos(x+class="stub"π
4
)sinclass="stub"π
4
=class="stub"7
10
2

f(x)=2sin2x=class="stub"49
25

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