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Solve the equation (sqrt(3))/2sinx-cosx=...

Solve the equation `(sqrt(3))/2sinx-cosx=cos^2x`

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`sqrt3/2sinx=cos^2x+cosx`
`sqrt3sinx=2(cos^2x+cosx)`
`3sin^2x=4(cos^2x+cosx)^2`
`3(1-cos^2x)=4(cos^4x+2cos^3x+cos^2x)`
`4cos^4x+8cos^3x+7cos^2x-3=0`
`(cosx+1)(2cosx-1)(2cos^2x+3cosx+3)=0`
`cosx=-1,1/2`
`x=(2n+1)pi,2npi+pi/3`.
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Knowledge Check

  • alpha is a root of the equation (2sinx-cosx)(1+cosx)=sin^2x beta is a root of the equation 3cos^2x-10cosx+3=0 and gamma is a root of the equation 1-sin2x=cosx-sinx where 0 le alpha, beta, gamma le pi//2 Answer the following question on above passage: sin(alpha-beta)

    A
    1
    B
    0
    C
    `(1-2sqrt6)/6`
    D
    `(sqrt3-2sqrt2)/6`

  • alpha is a root of the equation (2sinx-cosx)(1+cosx)=sin^2x beta is a root of the equation 3cos^2x-10cosx+3=0 and gamma is a root of the equation 1-sin2x=cosx-sinx where 0 le alpha, beta, gamma le pi//2 Answer the following question on above passage: sinalpha+sinbeta+singamma can be equal

    A
    `(14+3sqrt2)/(6sqrt2)`
    B
    44322
    C
    `(3+4sqrt2)/6`
    D
    `(1+sqrt2)/6`

  • The general solution of the equation sinx-3sin2x+sin3x=cosx-3cos2x+cos3x is

    A
    `npi+pi/8, n in I`
    B
    `(npi)/2+pi/8, n in I`
    C
    `(-1)^n((npi)/2)+pi/8, n in I`
    D
    `2npi+cos^-1(3/2), n in I`

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    Solve the equation 2(cos x+cos2x)+sin2x(1+2cos x)=2sinx for (-pilt=xlt=pi)

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