在锐角三角形中角ABC的对边分别为a,b,c已知(b^2+c^2-a^2)tanA=根号3bc若a=2 在锐角△ABC中,角A/B/C的对边分别为a、b、c,已知(...
\u5728\u9510\u89d2\u25b3ABC\u4e2d\uff0c\u89d2A\u3001B\u3001C\u7684\u5bf9\u8fb9\u5206\u522b\u4e3aa\u3001b\u3001c\uff0c\u5df2\u77e5(b2+c2-a2)tanA=3bc\uff0e\uff08\u2160\uff09\u6c42\u89d2A\uff1b\uff08\u2161\uff09\u82e5a=2\uff0c\u6c42\uff08I\uff09\u7531\u5df2\u77e5\u5f97b2+c2-a22bc?sinAcosA=32?sinA32\u53c8\u5728\u9510\u89d2\u25b3ABC\u4e2d\uff0c\u6240\u4ee5A=60\u00b0\uff0c\uff08II\uff09\u56e0\u4e3aa=2\uff0cA=60\u00b0\u6240\u4ee5b2+c2=bc+4\uff0cS=12bcsinA=34bc\u800cb2+c2\u22652bc?bc+4\u22652bc?bc\u22644\u53c8S=12bcsinA=34bc\u226434\u00d74=3\u6240\u4ee5\u25b3ABC\u9762\u79efS\u7684\u6700\u5927\u503c\u7b49\u4e8e3
1. (b+c-a)tanA=\u221a3bc (b+c-a)/(2bc)=(\u221a3/2)/tanA=(\u221a3/2)cosA/sinA \u7531\u4f59\u5f26\u5b9a\u7406\u5f97 cosA=(b+c-a)/(2bc) cosA=(\u221a3/2)cosA/sinA cosA[1-(\u221a3/2)/sinA]=0 A\u4e3a\u4e09\u89d2\u5f62\u5185\u89d2\uff0ccosA\u22600 1-(\u221a3/2)/sinA=0 sinA=\u221a3/2 \u4e09\u89d2\u5f62\u4e3a\u9510\u89d2\u4e09\u89d2\u5f62\uff0cA=\u03c0/3 2. S\u25b3ABC=(1/2)bcsinA \u7531\u5747\u503c\u4e0d\u7b49\u5f0f\u5f97\uff0c\u5f53b=c\u65f6\uff0cbc\u53d6\u5f97\u6700\u5927\u503c\uff0c\u6b64\u65f6B=C\uff0c\u53c8A=\u03c0/3 B=C=(\u03c0- \u03c0/3)/2=\u03c0/3=A b=c=a=2 \u6b64\u65f6\u4e09\u89d2\u5f62\u9762\u79ef\u6709\u6700\u5927\u503c(S\u25b3ABC)max=(1/2)\u00d72\u00d72\u00d7(\u221a3/2)=\u221a3
由余弦定理b²+c²-a²=2bc*cosA2bc*cosAtanA=2bcsinA=√3bc
sinA=√3/2 锐角三角形 A=π/3
a²=b²+c²-2bc*cosA=4 (b-c)²+bc=4 b=c时最大bc=4
S=1/2bcsinA=1/2*4*√3/2=√3
由余弦定理b²+c²-a²=2bc*cosA
2bc*cosAtanA=2bcsinA=√3bc
sinA=√3/2
A=π/3
4=a²=b²+c²-2bc*cosA≥2bc-2bc*cosA≥2bc(1-1/2)=bc
bc≤4
三角形面积S=bcsinA/2≤4*√3/2/2=√3
面积最大√3
bb+cc-aa=2bscosA
所以A=60°。
为等边三角形时面积最大，计算略
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