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Trig equation involving multiple angles

  1. Apr 21, 2013 #1

    Find all the values of ∏ in the interval 0 < θ < 2∏ for which
    2cos2 θ – sin θ = 1,
    giving each answer in terms of ∏

    I think the question is asking me to find the values of θ in terms of ∏ for all values between 0 < θ < 2∏

    This is what I have done so far
    I have worked out this question to give me a quadratic in terms of sin :

    2sin2 θ + sin θ – 1 = 0

    sinθ = 1/2 or -1

    so θ = ∏/6 or -∏/2

    3. The attempt at a solution

    Now, I have to find the values of ∏ within a range.
    The method I was taught at school was
    PV2 = 2∏ - PV1

    Which gave me: 5∏/6 and 3∏/2

    For the rest of the values, ± ∏

    So I did that giving me 11∏/6, 5∏/2

    Note: -∏/2 isn't part of the solution as it is out of the range

    So I think the final solutions are: ∏/6, 5∏/6, 3∏/2, 11∏/6, 5∏/2

    but my markscheme is saying that the solutions are ∏/6, 5∏/6, 3∏/2

    so I don't know what I have done wrong? Why should I not ±∏ from my PV 1 and 2 on this occasion?
    Am I doing something wrong?

    Note: I know there is some CAST method, but I haven't been taught this so I don't know how to use it.

    Thank you so much for your time and help!
    I really appreciate it! :)
    Last edited: Apr 21, 2013
  2. jcsd
  3. Apr 21, 2013 #2
    ##\displaystyle \forall \theta, \pi=4\int_{0}^{1} \sqrt{1-x^2}dx##. Clarify your question, or it will not be answered.
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