1)
sin2x = 3sinx*cos²x ;
sin2x =(3/2)*2sinx*cosx *cosx ; * * * 2sinx*cosx =sin2x * * *
sin2x =(3/2)*sin2x*cosx ;
2sin2x = 3sin2x*cosx ;
2sin2x -3sin2x*cosx = 0 ;
3sin2x( 2/3 - cosx) =0 ;
[ sin2x =0 ; [ 2x =πk , k ∈ Z
[ 2/3 -cosx =0 . [ cosx = 2/3.
[ x =(π/2)*k , k ∈ Z ;
[ x = ± arccos(2/3) +2πn , n∈ Z .
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2)
sin4x= sin2x * * * sin4x =sin2*2x = 2sin2x*cos2x * * *
2sin2x*cos2x =sin2x ;
2sin2x*cos2x - sin2x =0 ;
2sin2x(cos2x - 1/2) =0 ;
[ sin2x = 0 ; [ 2x =πk , k ∈ Z ;
[cos2x =1/2 . [ 2x =±π/3 +2πn , n∈Z.
[ x =(π/2)*k , k ∈ Z ;
[ x =±π/6 +πn , n∈Z.
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3)
cos2x+cos²x =0 ;
* * * cos2x =cos²x - sin²x =cos²x -(1 -cos²x) =2cos²x -1⇒cos²x =(1+cos2x)/2 * *
cos2x+(1+cos2x)/2 =0 ;
2cos2x +1 +cos2x =0 ;
3cos2x = -1 ;
cos2x = -(1/3) ;
2x =±(π -arccos(1/3)) +2πk , k ∈Z.
x =±0,5(π -arccos(1/3)) +πk , k ∈Z.
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4)
sin2x =cos²x ;
2sinx*cosx = cos²x ;
2sinx*cosx - cos²x =0 ;
cosx(2sinx -cosx) =0 ;
[ cosx =0 ; [ x =π/2 +πk , k∈Z; [x=π/2 +πk , k∈Z;
[2sinx - cosx =0 . [ 2sinx = cosx . [ tgx = 1/2 .
[x=π/2 +πk , k∈Z;
[x = arctg(1/2) +πn , n∈Z .