在三角形ABC中，a,b,c分别是∠A,B,C所对的边，已知a=根号3，b=3，∠C=30度，则∠A等于多少 在三角形ABC中，角A,B,C所对的边分别为a,b,c,已知...
\u5728\u25b3ABC\u4e2d,\u89d2A,B,C\u6240\u5bf9\u7684\u8fb9\u5206\u522b\u4e3aa,b,c,\u5df2\u77e5A=\u03c0/3,COSB=\u6839\u53f76/3\uff0c\u4e14c^2=a^2+(\u6839\u53f76 -1)b\uff0c\u6c42\u8fb9b\u7684\u957f\u7b54\uff1a
\u4e09\u89d2\u5f62ABC\u4e2d\uff0cA=\u03c0/3\uff0ccosB=\u221a6/3\uff1bc^2=a^2+(\u221a6-1)b
\u6839\u636e\u4e09\u89d2\u51fd\u6570\u57fa\u672c\u516c\u5f0f\u6c42\u5f97\uff1asinB=\u221a3/3
cosC=-cos(A+B)
=-cosAcosB+sinAsinB
=-(1/2)*(\u221a6/3)+(\u221a3/2)*(\u221a3/3)
=(3-\u221a6)/6
\u6839\u636e\u6b63\u5f26\u5b9a\u7406\uff1a
a/sinA=b/sinB=c/sinC=2R=\u221a3b
a=\u221a3b*sin(\u03c0/3)=3b/2
\u6839\u636e\u4f59\u5f26\u5b9a\u7406\u6709\uff1a
cosC=(a^2+b^2-c^2)/(2ab)
=[ (1-\u221a6)b+b^2 ]/(3b^2)
=(1-\u221a6+b)/(3b)
=(3-\u221a6)/6
\u6240\u4ee5\uff1a
6-6\u221a6+6b=9b-3\u221a6b
\u6240\u4ee5\uff1a
(3\u221a6-3)b=6(\u221a6-1)
b=2
sinB=\u6839\u53f73/3 cosB=\u221a6/3
A=2B
cosA=cos2B=1-2sin^2B=1-2/3=1/3
sinA=2\u221a2/3
\u6b63\u5f26\u5b9a\u7406a/sinA=b/sinB
a*\u221a3/3=2*2\u221a2/3 a=4\u221a6/3
sinC=sin(A+B)=sinAcosB+cosAsinB=4\u221a3/9+\u221a3/9=5\u221a3/9
c/sinC=b/sinB
c*\u221a3/3=2*5\u221a3/9
c=10/3
cosC=(a²+b²-c²)/2ab
将a=√3，b=3，C=30°代入上式，
√3/2=(3+9-c²)/6√3
解之得，c=√3=a
所以，三角形ABC是等腰三角形，∠C＝∠A＝30°
【另外】正弦定理
c/sinC=b/sinB=a/sinA
将a=c=√3，C=30°
√3/sin30°=√3/sinA
解之A＝30°或A=150°（舍去）
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