решите за 30 баллов ( с объяснением) sin(x+30)+cos(x+60)=1+cos2x. Created by shorokhova21. algebra-ru

решите за 30 баллов ( с объяснением) sin(x+30)+cos(x+60)=1+cos2x...

shorokhova21 @shorokhova21

July 2022 1 5 Report

решите за 30 баллов ( с объяснением) sin(x+30)+cos(x+60)=1+cos2x

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Task/27400429
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Решите sin(x+30)+cos(x+60 ) =1+cos2x
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cos(x+60°)+sin(x+30°) =1+cos2x ;
1 способ
cosx*cos60° - sinx*sin60° +sinx*cos30° +cosx*sin30° =1+cos2x ;
(1/2)*cosx - (√3 /2 )sinx + sinx* (√3 /2 ) +cosx*(1/2) =2cos²x ;
cosx = 2cos²x ;
2cosx (cosx -1/2)= 0  ;
cosx =0 ⇒ x =π/2+πn , n ∈Z .
или
cosx -1/2=0 ⇔cosx =1/2 ⇒ x = ±π/3 +2πk , k  ∈ Z.

ответ : π/2+πn ,n ∈Z ;  ±π/3 +2πk , k  ∈ Z.
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2 способ
cos(x+60°)+ cos(90° -(x+30°) ) =1+cos2x ;
cos(x+60°)  +cos(60°- x) =1+cos2x ;
2cos60°*cosx =2cos²x ;
cosx = 2cos²x ;
... дальше как в 1 способе
* * * * * * * P.S. * * * * * * *
cos(α+β) =cosαcosβ - sinαsinβ ;
sin(α+β) =sinαcosβ + cosαsinβ ;
cos2x =cos²x -sin²x = 2cos²x - 1⇒1+cos2x =2cos²x ;.
cos(90° - α) =sinα
cosα+cosβ= 2cos(α+β)/2 *cos(α-β)/2 .

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