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数学
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在△ABC中,三内角A、B、C及其对边a、b、c,满足a
2
-b
2
=
3
bc,sinC=
3
sinB
(Ⅰ)求角C的大小
(Ⅱ)若c=6,求△ABC面积.
人气:342 ℃ 时间:2019-08-20 04:08:54
解答
(Ⅰ)∵在△ABC中,sinC=
3
sinB,∴根据正弦定理,得c=
3
b
又∵a
2
-b
2
=
3
bc,∴a
2
-b
2
=3b
2
,解之得a=2b
∴△ABC中,a:b:c=2:1:
3
,可得a
2
=b
2
+c
2
△ABC是以a为斜边的直角三角形,
∵sinC=
c
a
=
3
2
,∴C=60° …(5分)
(Ⅱ)由(I)得a:b:c=2:1:
3
,
∴根据c=6,得b=2
3
∴Rt△ABC面积S=
1
2
bc=
1
2
×6×2
3
=6
3
…(9分)
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