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How to solve this trig equation

  1. Aug 22, 2012 #1


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    1. The problem statement, all variables and given/known data

    Solve this equation:


    For [itex]x \in [-\pi;\pi][/itex]

    2. Relevant equations


    3. The attempt at a solution

    [itex]cos^2(2x)=0,36 \Leftrightarrow cos(2x)=\sqrt{0,36} \Leftrightarrow 2x=cos^{-1}(\sqrt{0,36}) [/itex]

    And then I am not sure exactly how to proceed... When should I put in the [itex] 2p \pi [/itex] where [itex] x \in Z [/itex], to get all of the possible solutions?
    Last edited: Aug 22, 2012
  2. jcsd
  3. Aug 22, 2012 #2


    Staff: Mentor

    Not true. cos(2x) can also be negative. In your second equation, you took the square root of the right side, but not the left side.
    Also, you should simplify √(.36).

  4. Aug 22, 2012 #3


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    I corrected the mistake about not taking the square root on either side. So you mean I should put ± in front of the square root?
  5. Aug 22, 2012 #4


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    Staff Emeritus
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    Yes, use the ± .
  6. Aug 22, 2012 #5


    Staff: Mentor

    The domain for x is restricted to [##-\pi, \pi##], so you're going to get only a handful of solutions.
  7. Aug 23, 2012 #6


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    Ok I have come up with this solution:

    [itex]\frac{cos^{-1}(\pm \sqrt{0,36})}{2}+p\pi[/itex]

    Where the solutions are: [itex]cos^{-1}(\sqrt{0,36})-\pi, cos^{-1}(-\sqrt{0,36}), cos^{-1}(\sqrt{0,36}), cos^{-1}(-\sqrt{0,36})+\pi[/itex]

    Since the solutions have to be in the interval of -pi to pi.
  8. Aug 23, 2012 #7


    Staff: Mentor

    Why do you keep writing √(.36)? That simplifies to an exact value. What is this value?

    I think you would be better off by NOT using cos-1, since that will give you only one value. I would sketch a graph of y = cos(2x) on the interval [##-2\pi, 2\pi##] (since x ##\in## [##-\pi, \pi##]), and identify all of the points at which cos(2x) = ±B, where B is the simplified value of √(.36).

    EDIT: Also, your work above suggests that there are four solutions. I get quite a few more than that.
    Last edited: Aug 23, 2012
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