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Math Help - Solving Quadratic Trig Equations Algebraically

  1. #1
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    Solving Quadratic Trig Equations Algebraically

    Solve the following quadratic trig equations algebraically (without graphing technology) for the domain 0 < x < 360. Use exact solutions whenever possible.

    6cos^2(x) + cos(x) - 1 = 0


    I have done the following:

    6cos^2(x) + cos(x) - 1 = 0
    6cos^2(x) + 3cos(x) - 2cos(x) - 1 = 0
    3cos(x) [2cos(x) + 1] -1 [2cos(x) +1] = 0
    [3cos(x) -1] [2cos(x) + 1] = 0

    3cos(x) - 1 = 0
    3cos(x) = 1
    cos(x) = 1/3

    Quadrants I and IV
    Reference Angle: ???

    At this point I think I've done something wrong because there is no reference angle for cos(x) = 1/3.


    2cos(x) + 1 = 0
    2cos(x) = -1
    cos(x) = -1/2

    Quadrants II and III
    Reference Angle= 60 degrees

    180 - 60 = 120 degrees
    180 + 60 = 240 degrees



    Therefore, so far, I have solved for the values: 120 degrees and 240 degrees
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  2. #2
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    Quote Originally Posted by starshine84 View Post
    Solve the following quadratic trig equations algebraically (without graphing technology) for the domain 0 < x < 360. Use exact solutions whenever possible.

    6cos^2(x) + cos(x) - 1 = 0


    I have done the following:

    6cos^2(x) + cos(x) - 1 = 0
    6cos^2(x) + 3cos(x) - 2cos(x) - 1 = 0
    3cos(x) [2cos(x) + 1] -1 [2cos(x) +1] = 0
    [3cos(x) -1] [2cos(x) + 1] = 0

    3cos(x) - 1 = 0
    3cos(x) = 1
    cos(x) = 1/3

    Quadrants I and IV
    Reference Angle: ???

    At this point I think I've done something wrong because there is no reference angle for cos(x) = 1/3.
    Why not? 1/3 is between -1 and 1, of course, there is a reference angle. Any calculator will tell you that cos(70.53)= 1/3 (approximately). The problem did say "use exact solutions whenever possible.
    Or, if you prefer, exact solutions would be cos^{-1}(1/3) and 360- cos^{-1}(1/3).


    2cos(x) + 1 = 0
    2cos(x) = -1
    cos(x) = -1/2

    Quadrants II and III
    Reference Angle= 60 degrees

    180 - 60 = 120 degrees
    180 + 60 = 240 degrees



    Therefore, so far, I have solved for the values: 120 degrees and 240 degrees
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  3. #3
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    So are my final solutions x= 70.53, 120, 240, and 289.47??
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  4. #4
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    Thank you very much for you help!
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