Решить уравнение: sin^2(2x)+sin^2(3x)+sin^2(4x)+sin^2(5x)=2. Created by kz1501. algebra-ru

Решить уравнение: sin^2(2x)+sin^2(3x)+sin^2(4x)+sin^2(5x)=2...

kz1501 @kz1501

August 2022 1 13 Report

Решить уравнение:
sin^2(2x)+sin^2(3x)+sin^2(4x)+sin^2(5x)=2

Answers & Comments

oganesbagoyan

Verified answer

Sin²2x+sin²3x +sin²4x +sin²5x =2 ;
* * *cos2α =cos²α - sin²α = 1 -2sin²α ⇒2sin²α =1-cos2α ; sin²α =(1-cos2α)/2* * *
* * * (1-cos4x)/2 +(1-cos6x)/2 +(1-cos8x)/2 +(1-cos10x) =2;* * *
* * * удобно сначала уравнение умножить на 2 * * *
2sin²2x+2sin²3x +2sin²4x +2sin²5x =2*2 ;
(1-cos4x) +(1-cos6x) +(1-cos8x) +(1-cos10x) =4 ;
cos4x+cos6x +cos8x +cos10x =0 ;
cos6x+cos4x +cos10x +cos8x =0 ; * * * cosα +cosβ =2cos(α+β)/2 * cos(α -β)/2 * * *
2cos5xcosx +2cos9x*cosx =0 ;
2cosx(cos9x+cos5x) =0 ;
4cosx*cos2x*cos7x =0 ;
[cosx =0 ; cos2x =0; cos7x =0.
cosx =0 ⇒ x =π/2+π*n ,n∈Z.
cos2x =0 ⇒2x =π/2+π*n ⇔x =π/4+π/2*n ,n∈Z .
cos7x =0 ⇒7x =π/2+π*n ⇔x =π/14+π/7*n ,n∈Z.

ответ : π/2+π*n , π/4+π/2*n , π/14+π/7*n , n∈Z.

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