已知△ABC,∠B=60°,且sinA-sinC+22cos(A-C)=22.求sinC的值.-数学

题目简介

已知△ABC,∠B=60°,且sinA-sinC+22cos(A-C)=22.求sinC的值.-数学

题目详情

已知△ABC,∠B=60°,且sinA-sinC+
2
2
cos(A-C)=
2
2
.求sinC的值.
题型:解答题难度:中档来源:不详

答案

∵B=60°,∴∠A+C=120°
sinA-sinC=2cosclass="stub"A+C
2
sinclass="stub"A-C
2
=sinclass="stub"A-C
2

sinclass="stub"A-C
2
+
2
2
(1-2sin2class="stub"A-C
2
)=
2
2

sinclass="stub"A-C
2
(1-
2
sinclass="stub"A-C
2
)=0

sinclass="stub"A-C
2
=0或1-
2
sinclass="stub"A-C
2
=0

又∵0°<A<120°或0°<C<120°
-60°<class="stub"A-C
2
<60°

class="stub"A-C
2
=0°
class="stub"A-C
2
=45°

∴A=C=60° 或A=105°C=15°
当C=60°时,sin60°=
3
2

当C=15°时,sin15°=sin(45°-30°)=sin45°cos30°-cos45°sin30° =
6
-
2
4

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