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Math Help - Trigonometry Help

  1. #1
    Junior Member
    Joined
    Sep 2005
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    62

    Don't know how to solve these questions..

    Hi I am just sooo confused and I don't know what to do... or even start Can someone please explain this.

    The smallest positive number for which

    3sin(2x-6)=1

    is X= ______


    and


    3sin(2x-5)=1


    and


    4cos^2 x-9cosx+2=0


    I really don't know how to do these.
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    " The smallest positive number for which
    3sin(2x-6)=1
    is X= ______

    and
    3sin(2x-5)=1

    and
    4cos^2 x-9cosx+2=0 "

    x = a number?
    x is part of an angle or is an angle. Angles are in degrees and radians. Degrees are not "numbers", radians are "numbers", so we solve for x in radians.

    3sin(2x -6) = 1
    sin(2x -6) = 1/3 --------positive sine, so angle (2x -6) is in the 1st or 2nd quadrants.
    2x -6 = arcsin(1/3)
    2x -6 = 0.339836909 in the 1st quadrant, or (pi -0.339836909) = 2.801755744 in the 2nd quadrant.

    2x -6 = 0.339836909
    2x = 0.339836909 +6
    x = 6.339836909/2 = 3.169918455

    2x -6 = 2.801755744
    2x = 8.801755744
    x = 4.400877872 ----larger than 3.169918455

    Therefore, x = 3.169918455 is the smallest number to make 3sin(2x-6)=1 true. ----answer.

    ---------------------
    For 3sin(2x -5) = 1, follow the above and you should arrive at
    x = 5.339836909/2 = 2.669918455 -----answer.

    --------------------
    4cos^2(x) -9cos(x) +2 = 0
    That is a quadratic equation in cos(x). For simplicity, let y = cos(x),
    4y^2 -9y +2 = 0
    (4y -1)(y -2) = 0

    4y -1 = 0
    y = 1/4
    Or, cos(x) = 1/4 ---positive cosine, so angle x is in the 1st or 4th quadrants.
    x = arccos(1/4) = 1.318116072 in the 1st quadrant, or (2pi -1.318116072) = 4.965069235 in the 4th quadrant.

    y -2 = 0
    y = 2
    or, cos(x) = 2 ----reject this because maximum value of cosine of any angle is 1.0 only.

    Therefore, x = 1.318116702 is smallest number to make 4cos^2(x) -9cos(x) +2 = 0 true. ----answer.
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  3. #3
    Junior Member
    Joined
    Sep 2005
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    Thank you for the help
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