已知a,b为锐角,且cosa=3/5,sin(a-b)=5/13,求cosb的值

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已知a,b为锐角,且cosa=3/5,sin(a-b)=5/13,求cosb的值

已知a,b为锐角,且cosa=3/5,sin(a-b)=5/13,求cosb的值
已知a,b为锐角,且cosa=3/5,sin(a-b)=5/13,求cosb的值

已知a,b为锐角,且cosa=3/5,sin(a-b)=5/13,求cosb的值
∵a,b为锐角,cosa=3/5,sin(a-b)=5/13
∴sina=4/5,cos(a-b)=12/13
∴cosb=cos[(b-a)+a]=cos(b-a)cosa-sin(b-a)sina]=cos(a-b)cosa+sin(a-b)sina=(12/13)*(3/5)+(5/13)*(4/5)=56/65

a,b为锐角,cosa=3/5
所以可得:sina=√(1-cos^2a)=4/5
-π/2且有:cosb>0
当cos(a-b)=-12/13时有:
cosb=cos[a-(a-b)]
=cosacos(a-b)+sinasin(a-b)

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a,b为锐角,cosa=3/5
所以可得:sina=√(1-cos^2a)=4/5
-π/2且有:cosb>0
当cos(a-b)=-12/13时有:
cosb=cos[a-(a-b)]
=cosacos(a-b)+sinasin(a-b)
=-36/65+20/65
=-16/65<0 (舍去)
当cos(a-b)=12/13时有:
cosb=cos[a-(a-b)]
=cosacos(a-b)+sinasin(a-b)
=36/65+20/65
=56/65>0
综上可得:cosb=56/65

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