在锐角△ABC,角A、B、C所对的边分别为a,b,c,已知sinA=2√2/3,⑴求tan^B+C/2+sin^A/2的值,⑵若a=2,S△ABC=√2,求b的值?
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![在锐角△ABC,角A、B、C所对的边分别为a,b,c,已知sinA=2√2/3,⑴求tan^B+C/2+sin^A/2的值,⑵若a=2,S△ABC=√2,求b的值?](/uploads/image/z/5210428-4-8.jpg?t=%E5%9C%A8%E9%94%90%E8%A7%92%E2%96%B3ABC%2C%E8%A7%92A%E3%80%81B%E3%80%81C%E6%89%80%E5%AF%B9%E7%9A%84%E8%BE%B9%E5%88%86%E5%88%AB%E4%B8%BAa%2Cb%2Cc%2C%E5%B7%B2%E7%9F%A5sinA%3D2%E2%88%9A2%2F3%2C%E2%91%B4%E6%B1%82tan%EF%BC%BEB%2BC%2F2%2Bsin%EF%BC%BEA%2F2%E7%9A%84%E5%80%BC%2C%E2%91%B5%E8%8B%A5a%3D2%2CS%E2%96%B3ABC%3D%E2%88%9A2%2C%E6%B1%82b%E7%9A%84%E5%80%BC%3F)
在锐角△ABC,角A、B、C所对的边分别为a,b,c,已知sinA=2√2/3,⑴求tan^B+C/2+sin^A/2的值,⑵若a=2,S△ABC=√2,求b的值?
在锐角△ABC,角A、B、C所对的边分别为a,b,c,已知sinA=2√2/3,
⑴求tan^B+C/2+sin^A/2的值,⑵若a=2,S△ABC=√2,求b的值?
在锐角△ABC,角A、B、C所对的边分别为a,b,c,已知sinA=2√2/3,⑴求tan^B+C/2+sin^A/2的值,⑵若a=2,S△ABC=√2,求b的值?
sinA=2√2/3,
因为是锐角三角形,
所以cosA=1/3
tan^2[(B+C)/2]+sin^2 (A/2)
=tan^2(π-A)/2+sin^2(A/2)
=cot^2(A/2)+sin^2(A/2)
=(cos^2(A/2)/sin^2(A/2)+sin^2(A/2)
=[cos^2(A/2)+sin^2(A/2)*sin^2(A/2)]/sin^2(A/2)
=[cos^2(A/2)+(1-cos^2(A/2)*sin^2(A/2)]/sin^2(A/2)
=[cos^2(A/2)+sin^2(A/2)-cos^2(A/2)*sin^2(A/2)]/sin^2(A/2)
=[1-(1/4)sin^2A]/[(1-cosA)/2]
=[1-(1/4)2√2/3]/[(1-1/3)/2]
=(6-√2)/2
(2)
S=(1/2)bcsinA=(1/2)bc*2√2/3=√2
所以bc=3
根据余弦定理
a^2-b^2-c^2+2bccosA=0,
即4-b^2-c^2+2bc*1/3=0
解得:b=c=√3