三角形abc中,角A,B,C的对边分别为a,b,c,已知sinAsinB＋sinBsinC＋cos2

三角形abc中,角A,B,C的对边分别为a,b,c,已知sinAsinB＋sinBsinC＋cos2

证明:sinAsinB＋sinBsinC＋cos2B＝1sinAsinB＋sinBsinC＝1-cos2BsinAsinB＋sinBsinC＝ 2sin²B得sinA + sinC = 2 sinB因为正弦定理 a/sinA = b/sinB = c/sinC正弦函数和对应边成比例,即得a+c =2b移项 c-b = b-a所以a,b,c成等差数列（2）若c＝90°即a²+b²=c²c-b = b-ac²-2bc+b² = b²-2ab+a²c²-2bc+b² = c²-2abb²=2b(c-a)b=2c-2a由a+c =2b,可得4b=2a+2c5b=4c,b=4/5ca=3/5ca/b=3/4======以下答案可供参考======供参考答案1:（1）由于sinAsinB＋sinBsinC＋cos2B＝1，sinAsinB＋sinBsinC=2sinBsinB，sinA+sinC=2sinB。根据正玄定理，a+c=2b。证毕。（2）因为c＝90°，a*a+b*b=c*c,且a+c=2b。消c。得a/b=0.75.供参考答案2:（1）sinAsinB＋sinBsinC＋cos2B＝1 得到sinAsinB＋sinBsinC=1-cos2B=2sinB平方两边同时除以sinB平方 得到(sinA/sinB)+(sinC/sinb)=2 利用a/sinA=b/sinB=c/sinC得到（a/b）+（c/b）=2 进一步得到 a+c=2b 所以。。（2）因为角c=90度，则由勾股定理得到a^2+b^2=c^ 加上a+c=2b 推出c=2b-a 带入勾股定理 得到：a/b=3/4 。望采纳供参考答案3:sinAsinB＋sinBsinC＋cos2B＝1sinAsinB＋sinBsinC＝1-cos2BsinAsinB＋sinBsinC＝2sin²B根据正弦定理有 ab+bc=2b²就是 a+c=2b所以 a,b,c成等差数列。如果C=90度设 a=x-n,b=x, c=x+n则 (x-n)²+x²=(x+n)²可得 x=4na/b=(x-n)/x=3n/4n=3/4供参考答案4:a/sinA = b/sinB = c/sinC = 2RsinA = a/(2R), sinB = b/(2R), sinC = c/(2R)sinAsinB + sinBsinC + cos2B = 1a/(2R)*b/(2R) + b/(2R) *c/(2R) + 1 - 2(sinB)^2 = 1(a*b)/(4R^2) + (b*c)/(4*R^2) = 2 (sinB)^2 = 2*b^2/(4R^2)所以，a*b + b*c = 2*b^2a + c = 2b所以，a、b、c 成等差数列因为 C = 90°，则 sinC = 1，c = 2R，sinA = cosB(a+c)/(sinA+1) = 2b/(sinA+1) = b/sinB2sinB = sinA + 14(sinB)^2 = 4 - 4*(cosB)^2 = (sinA)^2 + 2sinA +1 = (cosB)^2 + 2cosB + 15(cosB)^2 + 2cosB - 3 = 0所以，cosB = sinA = [-2+√(2^2 + 4*5*3)]/(2*5) = [-2+8]/10 = 0.6s

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