已知0<A<π,且满足sinA+cosA=713,则5sinA+4cosA15sinA-7cosA=______.-数学

题目简介

已知0<A<π,且满足sinA+cosA=713,则5sinA+4cosA15sinA-7cosA=______.-数学

题目详情

已知0<A<π,且满足sinA+cosA=
7
13
,则
5sinA+4cosA
15sinA-7cosA
=______.
题型:填空题难度:中档来源:不详

答案

sinA+cosA=class="stub"7
13
两边平方得,2sinAcosA=-class="stub"120
169
<0,
∵0<A<π,∴class="stub"π
2
<A<π
,∴sinA-cosA>0
∴sinA-cosA=
1-2sinAcosA
=class="stub"17
13
,再由sinA+cosA=class="stub"7
13

解得,sinA=class="stub"12
13
,cosA=-class="stub"5
13

class="stub"5sinA+4cosA
15sinA-7cosA
=
5×class="stub"12
13
+4×(-class="stub"5
13
15×class="stub"12
13
-7×(-class="stub"5
13
=class="stub"8
43

故答案为:class="stub"8
43

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