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Answer 1 · 2018-02-25T14:31:38

#2pik , pik , (pik)/2#

Identities:

#color(red)bb(sin(A+B)=sinAcosB+cosAsinB)#

#color(red)bb(sin(2x)=2sinxcosx)#

#color(red)bb(cos(2x)=2cos^2x-1)#

#3sin(4theta)=3sin(2theta+2theta)->#

#=3(sin(2theta)cos(2theta)+cos(2theta)sin(2theta))=0#

#=6sin(2theta)cos(2theta)=0#

#=6(2sinthetacostheta(2cos^2x-1))=0#

#=6(2sinthetacosthetacos^2theta-2sinthetacostheta)=0#

#=(2sinthetacosthetacos^2theta-2sinthetacostheta)=0#

Factor:

#2sinthetacostheta(cos^2theta-1)=0#

#cos^2theta-1=0#

#cos^2theta=1#

#costheta=+-1#

#arccos(1)=>theta=2pik#

#arccos(-1)=>theta=pik#

#2sinthetacostheta=0#

#sin(2theta)=0#

#2theta=arcsin(sin(2theta))=arcsin(0)=>theta=(2pik)/2 , (pik)/2#

All solutions:

#2pik , pik , (pik)/2#

For:

#k in ZZ#