已知a,b,c分别是△ABC中角A,B,C的对边,且sin²A+sin²C-sin²B=sinAsinC(1)求角B的大小(2)若c=3a,求tanA的值
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![已知a,b,c分别是△ABC中角A,B,C的对边,且sin²A+sin²C-sin²B=sinAsinC(1)求角B的大小(2)若c=3a,求tanA的值](/uploads/image/z/1811929-49-9.jpg?t=%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%E5%88%86%E5%88%AB%E6%98%AF%E2%96%B3ABC%E4%B8%AD%E8%A7%92A%2CB%2CC%E7%9A%84%E5%AF%B9%E8%BE%B9%2C%E4%B8%94sin%26%23178%3BA%2Bsin%26%23178%3BC-sin%26%23178%3BB%3DsinAsinC%281%29%E6%B1%82%E8%A7%92B%E7%9A%84%E5%A4%A7%E5%B0%8F%EF%BC%882%EF%BC%89%E8%8B%A5c%3D3a%2C%E6%B1%82tanA%E7%9A%84%E5%80%BC)
已知a,b,c分别是△ABC中角A,B,C的对边,且sin²A+sin²C-sin²B=sinAsinC(1)求角B的大小(2)若c=3a,求tanA的值
已知a,b,c分别是△ABC中角A,B,C的对边,且sin²A+sin²C-sin²B=sinAsinC
(1)求角B的大小
(2)若c=3a,求tanA的值
已知a,b,c分别是△ABC中角A,B,C的对边,且sin²A+sin²C-sin²B=sinAsinC(1)求角B的大小(2)若c=3a,求tanA的值
sin²A+sin²C-sin²B=sinAsinC
根据正弦定理
a²+c²-b²=ac
b²=a²+c²-ac
根据余弦定理:
b²=a²+c²-2accosB
2cosB=1
cosB=1/2
B=π/3
(2)c=3a ( ∠C>∠A a<90°)
sinC=3sinA
sin²A+sin²C-sin²B=sinAsinC
sin²A+9sin²A-3/4=3sin²A
7sin²A=3/4
sin²A=3/28
cos²A=25/28
tan²A=3/25
tanA=√3/5
(1)1/2 -1/2 cos2A +1/2 -1/2 cos2C-sin^2 B=1/2 COS(A-C) -1/2 COS(A+C)
1-1/2 (COS2A+COS2C)-SIN^2B=1/2COS(A-C)-1/2COS(A+C)
1-1/2 *2COS(A+C)COS(A-C)-SIN^2B=1/2COS(A-C)-1/2COS(A+C)
COS^2B-COS(A...
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(1)1/2 -1/2 cos2A +1/2 -1/2 cos2C-sin^2 B=1/2 COS(A-C) -1/2 COS(A+C)
1-1/2 (COS2A+COS2C)-SIN^2B=1/2COS(A-C)-1/2COS(A+C)
1-1/2 *2COS(A+C)COS(A-C)-SIN^2B=1/2COS(A-C)-1/2COS(A+C)
COS^2B-COS(A+C)COS(A-C)=1/2COS(A-C)-1/2COS(A+C)
COS^2 B+COSBCOS(A-C)=1/2COS(A-C)+1/2COSB
COS(A-C)(COSB-1/2)=1/2COSB-COS^2B=COSB(1/2-COSB)
(COSB-1/2)(COS(A-C)+COSB)=0
COSB=1/2或cos(A-C)+COSB=0
所以B=60
(2)COSB=(a^2+c^2-b^2)/(2ac)=(10a^2-b^2)/(6a^2)=1/2
10a^2-b^2=3a^2
7a^2=b^2
a/sinA=b/sinB
sinA=1/根号7 *根号3/2
cosA=(b^2+c^2-a^2)/(2bc) =(7a^2+9a^2-a^2)/(2根号7 a *3a)=5/(2根号7)
tanA=sinA/cosA=根号3 /(2根号7) *(2根号7)/5= 根号3/5
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