在三角形ABC中,角A,B,C所对的边分别为a,b,c,且cosA=1/3.(1)求cos²[﹙B+C)/2]+cos2A的值 (2)若a=2,c=3/2,求角C的大小
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![在三角形ABC中,角A,B,C所对的边分别为a,b,c,且cosA=1/3.(1)求cos²[﹙B+C)/2]+cos2A的值 (2)若a=2,c=3/2,求角C的大小](/uploads/image/z/1601052-60-2.jpg?t=%E5%9C%A8%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E4%B8%AD%2C%E8%A7%92A%2CB%2CC%E6%89%80%E5%AF%B9%E7%9A%84%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa%2Cb%2Cc%2C%E4%B8%94cosA%3D1%2F3.%281%29%E6%B1%82cos%26%23178%3B%5B%EF%B9%99B%2BC%29%EF%BC%8F2%5D%2Bcos2A%E7%9A%84%E5%80%BC+%EF%BC%882%EF%BC%89%E8%8B%A5a%3D2%2Cc%3D3%EF%BC%8F2%2C%E6%B1%82%E8%A7%92C%E7%9A%84%E5%A4%A7%E5%B0%8F)
在三角形ABC中,角A,B,C所对的边分别为a,b,c,且cosA=1/3.(1)求cos²[﹙B+C)/2]+cos2A的值 (2)若a=2,c=3/2,求角C的大小
在三角形ABC中,角A,B,C所对的边分别为a,b,c,且cosA=1/3
.(1)求cos²[﹙B+C)/2]+cos2A的值 (2)若a=2,c=3/2,求角C的大小
在三角形ABC中,角A,B,C所对的边分别为a,b,c,且cosA=1/3.(1)求cos²[﹙B+C)/2]+cos2A的值 (2)若a=2,c=3/2,求角C的大小
⑴(B+C)/2=(180°-A)/2=90°-A/2,
∵cosA=2(cosA/2)^2-1,∴cosA/2=√6/3,∴sinA/2=√3/3
cos2A=2(cosA)^2-1=2×(1/3)^2-1=-7/9,
∴原式=(sinA/2)^2+cos2A=1/3-7/9=-4/9.
⑵∵1/2
1、 cos²[﹙B+C)/2]+cos2A
=cos²[(180°-A)/2]+cos2A
=cos²[90°-A/2]+2cos²A-1
=sin²A/2+2cos²A-1
=[√(1-cosA)/2]²+2cos²A-1
=[(1-cosA)/2]+2cos²A-...
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1、 cos²[﹙B+C)/2]+cos2A
=cos²[(180°-A)/2]+cos2A
=cos²[90°-A/2]+2cos²A-1
=sin²A/2+2cos²A-1
=[√(1-cosA)/2]²+2cos²A-1
=[(1-cosA)/2]+2cos²A-1
=[(1-1/3)/2+2×(1/3)²-1
=1/3+2/9-1
=5/9-1
=-4/9
2、cosA=1/3,
∴在△中sinA=√(1-1/9)=2√2/3
∴正弦定理:a/sinA=c/sinC
sinC=sinA×c/a=(2√2/3×3/2)/2=√3/2
∴C=60°或120°
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