已知ABC是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c已知A,B,C是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c,若cos(A-C)-cos(A+C)=3/2,且b的2次方=ac,求角B的值,若B=根号3,求三角形ABC的周长
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/09 05:02:25
![已知ABC是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c已知A,B,C是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c,若cos(A-C)-cos(A+C)=3/2,且b的2次方=ac,求角B的值,若B=根号3,求三角形ABC的周长](/uploads/image/z/2605937-41-7.jpg?t=%E5%B7%B2%E7%9F%A5ABC%E6%98%AF%E9%94%90%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E7%9A%84%E4%B8%89%E4%B8%AA%E5%86%85%E8%A7%92%2CA%2CB%2CC%E7%9A%84%E5%AF%B9%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa%2Cb%2Cc%E5%B7%B2%E7%9F%A5A%2CB%2CC%E6%98%AF%E9%94%90%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E7%9A%84%E4%B8%89%E4%B8%AA%E5%86%85%E8%A7%92%2CA%2CB%2CC%E7%9A%84%E5%AF%B9%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa%2Cb%2Cc%2C%E8%8B%A5cos%28A-C%29-cos%28A%2BC%29%3D3%2F2%2C%E4%B8%94b%E7%9A%842%E6%AC%A1%E6%96%B9%3Dac%2C%E6%B1%82%E8%A7%92B%E7%9A%84%E5%80%BC%2C%E8%8B%A5B%3D%E6%A0%B9%E5%8F%B73%2C%E6%B1%82%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E7%9A%84%E5%91%A8%E9%95%BF)
已知ABC是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c已知A,B,C是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c,若cos(A-C)-cos(A+C)=3/2,且b的2次方=ac,求角B的值,若B=根号3,求三角形ABC的周长
已知ABC是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c
已知A,B,C是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c,若cos(A-C)-cos(A+C)=3/2,且b的2次方=ac,求角B的值,若B=根号3,求三角形ABC的周长
已知ABC是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c已知A,B,C是锐角三角形ABC的三个内角,A,B,C的对边分别为a,b,c,若cos(A-C)-cos(A+C)=3/2,且b的2次方=ac,求角B的值,若B=根号3,求三角形ABC的周长
答:
(1)
cos(A-C)-cos(A+C)=3/2
2sinAsinC=3/2
sinAsinC=3/4
因为:b^2=ac,所以:a/b=b/c
根据正弦定理:a/sinA=b/sinB=c/sinC
所以:a/b=sinA/sinB=b/c=sinB/sinC
(sinB)^2=sinAsinC=3/4
sinB=√3/2
B=60°或者B=120°(不符合锐角三角形需舍去)
因为三角形ABC是锐角三角形,所以B=60°
(2)
sinAsinC
=sinAsin(180°-60°-A)
=sinA(sin120°cosA-cos120°sinA)
=√3sinAcosA/2+sinAsinA/2=3/4
所以:√3sin2A-cos2A=2
结合:(sin2A)^2+(cos2A)^2=1解得:
sin2A=√3/2,cos2A=-1/2
2A=120°,A=60°
结合(1)知道:A=B=C=60°
所以a=b=c=√3
周长为3√3