Решите уравнение:cos 2x +3sin2x=3Кто напишит подробное решение, тому лучший ответ!

Решите уравнение:

cos 2x +3sin2x=3

Кто напишит подробное решение, тому лучший ответ!

  • cos2x+3sin2x=3

    Попробуем открыть по формуле cos2x=cos^2x-sin^2x подставим

      cos^2x-sin^2x+3sin2x=3

    опять    sin2x откроем по формуле  sin2x=2sinx*cosx подставим

     cos^2x-sin^2x +3(2sinx*cosx)=3

    cos^2x-sin^2x+6sinx*cosx =3

    вспомним что cos^2x=1-sin^2x  подставим

    1-sin^2x-sin^2x+6V(1-sin^2x)*sinx=3

    1-2sin^2x+6V(1-sin^2x)*sinx =3

    -2sin^2x+6V(1-sin^2x) *sinx=2

    поделим на 2

    -sin^2x+3V(1-sin^2x)*sinx=1  

    3V(1-sin^2x)*sinx=1+sin^2x

    можно заменить sinx=t тогда

    3V(1-t^2)t=1+t^2

    возмедем обе части в квадрат

    9t^2(1-t^2)=1+2t^2+t^4

    9t^2-9t^4=1+2t^2+t^4

    t^4+9t^4+2t^2-9t^2+1 =0

    10t^4-7t^2+1=0

    биквдатратное уравнение опять заменим на t^2=a

    10a^2-7a+1=0

    D=49-4*10*1=V9=3

    a1=7+3/20=1/2

    a2=7-3/20=1/5

    a=t^2

    t^2=1/2

    t=V2/2

    t=1/5

    t=1/V5

    t=sinx

    sinx=V2/2

    x=pi/4

     sinx=1/V5

    x=-1arcsin(1/V5)+2pi*k 

     

     

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