ответ:
пошаговое объяснение:
sin105°*cos75° = sin(180° -75°)*cos75° = sin75°*cos75° =(sin2*75°)/2 =
(sin150°)/2 =(sin(180°- 30°))/2 = (sin30°)/2 =(1/2) /2 =1/4.
1) sin105°*sin75° = sin(180° -75°)*sin75° = sin75°*sin75° =sin²75°=
(1 -cos2*75°)/2 =(1 -cos150°)/2 = (1 -cos(180° -30°) )/2 = (1+cos30°) /2 =
(2+√3) / 4 .
* * * sin²75° =(sin45°cos30° + cos45°sin30°) ² = ( (1/√2)*(√3)/2 +(1/√2)*(1)/2) ) ² =(1/8) *(√3 +1) ² =(1/8) *(3 +2√3 +1)= (1/4) *(2 +√3 )= (2 +√3 ) /4.
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2) 4sin(π/6 -β)cos(π/6+β)= 4 *(sin(π/6 -β+π/6+β) + sin(π/6 -β-π/6-β) )/2 =
2 *(sin π/3 + sin( -2β) ) = 2 *( (√3)/2 - sin2β ) =√3 -2 sin2β.
* * * а если преобразование начнем с правой стороны равенства , то
3 - 4cos²β = 4(1 - cos²β) -1 =4sin²β -1 =2*2sin²β -1 =2(1 -cos2β) -1 =
2(1 - cos2β -1/2) = 2(1/2 -cos2β) = 2(cosπ/3 -cos2β) = 2(cosπ/3 -cos2β) =
- 4sin(π/6- β)*sin(π/6+ β) .
ответ:
пошаговое объяснение:
sin105°*cos75° = sin(180° -75°)*cos75° = sin75°*cos75° =(sin2*75°)/2 =
(sin150°)/2 =(sin(180°- 30°))/2 = (sin30°)/2 =(1/2) /2 =1/4.
1) sin105°*sin75° = sin(180° -75°)*sin75° = sin75°*sin75° =sin²75°=
(1 -cos2*75°)/2 =(1 -cos150°)/2 = (1 -cos(180° -30°) )/2 = (1+cos30°) /2 =
(2+√3) / 4 .
* * * sin²75° =(sin45°cos30° + cos45°sin30°) ² = ( (1/√2)*(√3)/2 +(1/√2)*(1)/2) ) ² =(1/8) *(√3 +1) ² =(1/8) *(3 +2√3 +1)= (1/4) *(2 +√3 )= (2 +√3 ) /4.
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2) 4sin(π/6 -β)cos(π/6+β)= 4 *(sin(π/6 -β+π/6+β) + sin(π/6 -β-π/6-β) )/2 =
2 *(sin π/3 + sin( -2β) ) = 2 *( (√3)/2 - sin2β ) =√3 -2 sin2β.
* * * а если преобразование начнем с правой стороны равенства , то
3 - 4cos²β = 4(1 - cos²β) -1 =4sin²β -1 =2*2sin²β -1 =2(1 -cos2β) -1 =
2(1 - cos2β -1/2) = 2(1/2 -cos2β) = 2(cosπ/3 -cos2β) = 2(cosπ/3 -cos2β) =
- 4sin(π/6- β)*sin(π/6+ β) .