若cos(a+b)sin(a-b)+1/2sinacosa=0,3sin^2 a+2sin^2,b=1,a,b都为锐角,求sin(a+b)
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若cos(a+b)sin(a-b)+1/2sinacosa=0,3sin^2 a+2sin^2,b=1,a,b都为锐角,求sin(a+b)
若cos(a+b)sin(a-b)+1/2sinacosa=0,3sin^2 a+2sin^2,b=1,a,b都为锐角,求sin(a+b)
若cos(a+b)sin(a-b)+1/2sinacosa=0,3sin^2 a+2sin^2,b=1,a,b都为锐角,求sin(a+b)
sin(2a)+sin(-2b)
= sin[(a-b)+(a+b)]+sin[(a-b)-(a+b)]
=sin(a-b)cos(a-b)+cos(a-b)sin(a+b)+sin(a-b)cos(a-b)-cos(a-b)sin(a+b)
=2sin(a-b)cos(a+b)
∴cos(a+b)sin(a-b)=1/2sin2a-1/2sin2b
∵cos(a+b)sin(a-b)+1/2sinacosa=0
∴1/2sin2a-1/2sin2b+1/4sin2a=0
∴3sin2a=2sin2b ①
∵3sin²a+2sin²b=1
∴3/2(1-cos2a)+1-cos2b=1
∴3cos2a=3-2cos2b ②
①²+②²:
9=4sin²2b+9-12cos2b+4cos²2b
∴12cos2b=4
∴cos2b=1/3, cos2a=7/9
∴1-2sin²b=1/3,sin²b=1/3,
∵b是锐角
∴sinb=√3/3,cosb=√6/3
1-2sin²a=7/9,sin²a=1/9
∵a是锐角
∴sina=1/3,cosa=2√2/3
∴sin(a+b)
=sinacosb+cosasinb
=1/3*√6/3+2√2/3*√3/3
=√6/3
cos(A+B)sin(A-B)+(1/2)sinAcosA=0,
∴(cosAcosB-sinAsinB)(sinAcosB-cosAsinB)+(1/2)sinAcosA=0,
∴（3/2）sinAcosA-sinBcosB=0,
(3/2)sin2A=sin2B,①
3sin^A+2sin^B=1,
∴(3/2)(1-cos2A)=cos2B，②
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cos(A+B)sin(A-B)+(1/2)sinAcosA=0,
∴(cosAcosB-sinAsinB)(sinAcosB-cosAsinB)+(1/2)sinAcosA=0,
∴（3/2）sinAcosA-sinBcosB=0,
(3/2)sin2A=sin2B,①
3sin^A+2sin^B=1,
∴(3/2)(1-cos2A)=cos2B，②
①^+②^,(9/4)(2-2cos2A)=1,cos2A=7/9,
代入②，cos2B=1/3,
A,B都是锐角，
∴sin2B=2√2/2，sin2A=4√2/9，
∴cos(2A+2B)=-1/2,
∴sin(A+B)=√{[1-cos(2A+2B)]/2}=√3/2.
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