Question

Au loc egalitatile?
1.sin(120 grade - x)=sin x+sin(60gr-x)
 2.cos x -cos (x+60 gr )+cos(x+120 gr)=0
3.$sin x^{2} (x+y)=sin x^{2} x+sin x^{2} y+2sinx*siny*cos(x+y)$

Answer 1

1)sin(120-x)=sin120*cosx-sinx*cos120
sin120=sin(60+60)=sin60*cos60+sin60*cos60=√3/2*1/2+√3/2*1/2=√3/4+√3/4=2√3/4=√3/2=>sin120=√3/2
cos120=cos(60+60)=cos60*cos60-sin60*sin60=1/2*1/2-√3/2*√3/2=1/4-3/4=-2/4=-1/2=>cos120=-1/2
Deci sin(120-x)=√3/2*cosx+1/2*sinx  (1)

sinx+sin(60-x)=sinx+sin60*cosx-sinx*cos60=sinx+√3/2cosx-1/2sinx=(2sinx-sinx)/2+√3/2*cosx=sinx/2+√3/2*cosx  (2)
Din (1) si (2) => ca egalitattea este adevarata
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2)cosx-(cosx*cos60-sinx*sin60)+cosx*cos120-sinx*sin120=0 =>
=> cosx-(1/2*cosx-√3/2*sinx)-1/2cosx-√3/2sinx=0 =>
=> cosx-1/2*cosx+√3/2*sinx -1/2cosx-√3/2sinx=o=>
=>2cosx-cosx/2 +√3/2sinx-1/2cosx-√3/2sinx=0=>
=> 1/2cosx-1/2cosx+√3/2sinx-√3/2simx=0 => 0=0 "A"
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