若tan(a+b)=2tan a,求证3sin b=sin(2a+b)_数学解答

若tan(a+b)=2tan a,求证3sin b=sin(2a+b)

人气：335 ℃　时间：2020-06-14 11:11:55

解答

tan(a+b)=2tana
sin(a+b)/cos(a+b)=2sina/cosa
sinacosbcosa+sinbcosacosa=2sinacosacosb-2sinasinasinb
sinb(cos^2a+2sin^2a)=sinacosacosb
2sinb(cos^2a+2(1-cos^2a)=sin(2a)cosb
sinb(4-2cos^2a)=sin(2a)cosb
sinb(3-2cos^2a+1)=sin(2a)cosb
sinb(3-cos(2a))=sin(2a)cosb
3sinb-sinbcos(2a)=sin(2a)cosb
3sinb=sinbcos(2a)+sin(2a)cosb
3sinb=sinb(2a+b)