已知在三角形ABC中,内角ABC所对的边分别是abc,且满足2asin(C+π/6)=b+c 求A的值
推荐于2017-11-22
展开全部
2asin(C+π/6) = b+c
根据正弦定理有:
2sinAsin(C+π/6) = sinB+sinC
2sinA{sinCcosπ/6+cosCsinπ/6) = sinB+sinC
sinA{√3sinC+cosC) = sinB+sinC
√3sinAsinC+sinAcosC = sinB+sinC
又,sinB=sin(A+C) = sinAcosC+cosAsinC
∴ √3sinAsinC+sinAcosC = sinAcosC+cosAsinC +sinC
∴ √3sinAsinC =cosAsinC +sinC
两边同除以sinC得:
√3sinA =cosA + 1
√3sinA - cosA = 1
2(sinAcosπ/6 - cosAsinπ/6) = 1
sin(A-π/6) = 1/2
A-π/6 ≠ 5π/6
∴ A-π/6 = π/6
∴ A = π/3
根据正弦定理有:
2sinAsin(C+π/6) = sinB+sinC
2sinA{sinCcosπ/6+cosCsinπ/6) = sinB+sinC
sinA{√3sinC+cosC) = sinB+sinC
√3sinAsinC+sinAcosC = sinB+sinC
又,sinB=sin(A+C) = sinAcosC+cosAsinC
∴ √3sinAsinC+sinAcosC = sinAcosC+cosAsinC +sinC
∴ √3sinAsinC =cosAsinC +sinC
两边同除以sinC得:
√3sinA =cosA + 1
√3sinA - cosA = 1
2(sinAcosπ/6 - cosAsinπ/6) = 1
sin(A-π/6) = 1/2
A-π/6 ≠ 5π/6
∴ A-π/6 = π/6
∴ A = π/3
推荐律师服务:
若未解决您的问题,请您详细描述您的问题,通过百度律临进行免费专业咨询