Sin^2(x)+sin^2(2x)+sin^2(3x)+sin^2(4x)=2 нужна - вопрос №708123822223242 от mtoropchin 15.11.2020 00:57

Question

Алгебра: Sin^2(x)+sin^2(2x)+sin^2(3x)+sin^2(4x)=2 нужна

mtoropchin · Answer

Sin²2x+sin²4x=(1-sin²x)+(1-sin²3x)sin²2x+sin²4x=cos²x+cos²3x(1-cos4x)/2+(1-cos8x)/2=(1+cos2x)/2+(1+cos6x)/21-cos4x+1-cos8x=1+cos2x+1+cos6xcos4x+cos8x+cos2x+cos6x=02cos6xcos2x+2cos4xcos2x=02cos2x(cos6x+cos4x)=02cos2x*2cosx*cos5x=0cos2x=0⇒2x=π/2+πn⇒x=π/4+πn/2,n∈zcosx=0⇒x=π/2+πn,n∈zcos5x=0⇒5x=π/2+πn⇒x=π/10+πn/5...