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March 30, 2017

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Find all solutions of 4(cos(x))**2-4 sin(x)-5=0 in the interval (6pi, 8pi)

I tried to work it out and got: 4(cos**2x)-4cosx -9 = 0, but I can't figure out what cosx = from there to finish the problem.

  • Trigonometry - ,

    ________________________________________

    Remark :

    ( cos x ) ^ 2 = 1 - ( sin x ) ^ 2
    ________________________________________


    4 ( cos x ) ^ 2 - 4 sin( x ) - 5 = 0

    4 [ 1 - ( sin x ) ^ 2 ] - 4 sin( x ) - 5 = 0

    4 - 4 ( sin x ) ^ 2 - 4 sin ( x ) - 5 = 0

    - 4 ( sin x ) ^ 2 - 4 sin ( x ) - 1 = 0 Multiply both sides by - 1

    4 ( sin x ) ^ 2 + 4 sin ( x ) + 1 = 0
    ________________________________________

    Remark :

    ( a + b ) ^ 2 = a ^ 2 + 2 a b + b ^ 2

    So :

    4 sin ( x ) ^ 2 + 4 sin ( x ) + 1 = [ 2 sin ( x ) + 1 ] ^ 2

    Becouse :

    [ 2 sin ( x ) + 1 ] ^ 2 =

    [ 2 sin ( x ) ] ^ 2 + 2 * 2 sin ( x ) + 1 ^ 2 =

    4 sin ( x ) ^ 2 + 4 sin ( x ) + 1


    4 ( sin x ) ^ 2 + 4 sin ( x ) + 1 = 0 is same :

    [ 2 sin ( x ) + 1 ] ^ 2 = 0
    ________________________________________

    [ 2 sin ( x ) + 1 ] ^ 2 = 0 Take the square root of both sides

    2 sin ( x ) + 1 = 0 Subtract 1 from both sides

    2 sin ( x ) = - 1 Divide bboth sides by 2

    sin ( x ) = - 1 / 2


    sin ( x ) = - 1 / 2 for :

    x = 7 pi / 6 = 210 °

    and

    x = 11 pi / 6 = 330 °


    The period of sin ( x ) is 2 pi

    So :

    sin ( x ) = - 1 / 2 for :

    x = 2 n pi +7 pi / 6

    and

    x = 2 n pi + 11 pi / 6

    n is an integer


    In the interval ( 6 pi, 8 pi )

    x = 2 * 3 pi + 7 pi / 6

    and

    x = 2 * 3 pi + 11 pi / 6

    Solutions :

    x = 6 pi + 7 pi / 6

    and

    x = 6 pi + 11 pi / 6


    OR

    x = 6 * 6 pi / 6 + 7 pi / 6 = 36 pi / 6 + 7 pi / 6 = 43 pi / 6

    and

    x = 6 * 6 pi / 6 + 11 pi / 6 = 36 pi / 6 + 11 pi / 6 = 47 pi / 6

  • Trigonometry - ,

    You apparently think that sinx = 1+cosx
    The correct identity is cos^2 + sin^2 = 1
    so, cos^2 x = 1-sin^2 x

    4(1-sin^2 x) - 4sinx - 5 = 0
    4 - 4sin^2 x - 4sinx - 5 = 0

    4sin^2 x + 4sinx + 1 = 0
    (2sinx + 1)^2 = 0
    2sinx + 1 = 0
    sinx = -1/2

    Take it from there.

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