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Thread: On a Problem

  1. #1 On a Problem 
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    Two weeks ago, I posed to myself the problem of determining all the possible values of x which satisfy the equation:


    I believe I have come up with the answer; however, I would appreciate someone running through my result, not least because my answer would have me contend with the existence of the cosine of an imaginary angle - which, while not exactly a problem as such, would seem to produce inconsistencies in my proof.

    Rather than attempt a laborious explanation of why I would like people to tell me if I'm going wrong somewhere, perhaps I should simply get on with explaining what I have done:

    1. I begin by assuming that the equation is satisfied for some value i.e.


    2. Now, it is quite clear from the above that



    and since (less than or equal to, I mean, not strictly less than)

    this implies



    for otherwise it would never cancel out with the cosine.

    3. Thus, I may now construct a quadratic equation , where , and a = b = 1.

    (The justification is quite simple: I have simply generalised the values of between the limits of 1 and -1, and sent the freewheeling constant to the right hand side of the equation)

    Following which I proceed to use the quadratic equation to determine my values of Q, solving separately for the cases when c = 1 and c = -1 to obtain my limits for the possible values of x. That gives me my answer, recorded separately below.

    Here is where my difficulties arise, from a rigorous standpoint. Is a variable here? For, if so, I have no right whatsoever to apply the quadratic equation to determine the value of Q. From an intuitive perspective, it would seem that Q must be a variable - however, my intuition has been wrong before, and so I seek clarification on this subtle point. I have assumed, you see, that there may be more than one possible value of Q which satisfies my original equation (giving it roughly the shape and flavour of a variable), which is much harder to prove, at least for me, and even then I am not sure if it would fit the definition of a variable.

    However, let us move on. My real problems arise with my answers. Solving the above quadratic for , I obtain the following limits:



    Now, here's the kicker. Quite evidently, my equation holds for imaginary values of x as well. However, how exactly can I then state that my cosine remains within the values of 1 and - 1 for imaginary values too? I did attempt to prove it, but it involved defining the cosine of an imaginary angle (not to mention defining the imaginary angle itself), and quite frankly there is every possibility that my definition may clash with established mathematical research - rendering any conceivable hope a frail thing indeed. That, then, is my plea: to assuage my doubts regarding the validity of my statement that the cosine of an angle is always between 1 and - 1, regardless of the angle's existence existence as complex or real.

    I thank anyone who's read this far. This is not homework, let me assure you, only a question I asked myself. Thank you yet again for devoting any time you may have to clarifying my difficulties.


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  3. #2  
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    The equation posted above is not a quadratic equation, and be trivially solved. Since c is itself a function of Q, taking c to be a constant fixes the value of Q automatically. Equations of this forms are known as transcendental equations and generally cannot e solved analytically except for trivial cases.

    Also, regarding complex angle: the do exists. In fact they are some times reffered to as hyperbolic angles. example, cos (i x) = cosh (x), cosh is called hyperbolic cosine function.


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  4. #3  
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    The equation posted above is not a quadratic equation, and be trivially solved. Since c is itself a function of Q, taking c to be a constant fixes the value of Q automatically. Equations of this forms are known as transcendental equations and generally cannot e solved analytically except for trivial cases.

    Also, regarding complex angle: the do exists. In fact they are some times referred to as hyperbolic angles. example, cos (i x) = cosh (x), cosh is called hyperbolic cosine function.
    Firstly, thank you for replying to my post. It is very highly appreciated.

    Secondly, then surely this means my approach is indeed correct? I have allowed c to vary, and solved only for the limits of c to state the limits of Q itself. To be precise, I have shown that the sum is allowed to vary between the limits of 1 and -1, and then showed the necessary values that Q must take in order to be between those limits. Rather than state every single value of Q that emerges between those limits (something which borders impossibility), I have only stated the end limits.

    For example, suppose . This will lead me to state, equivalently, that



    Quite clearly, this is a quadratic equation (for I certainly can't think of it to be anything else - it certainly fits the form of ). If I choose to solve this, I should definitely get an answer between the end limits I stated in my previous post. That is all I have done. How, then, can this not be a quadratic equation, putting aside Q's dubious claim to be a variable?

    And thank you for enlightening me to the existence to hyperbolic functions. I must look into them.
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  5. #4  
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    the issue, is, at the values that will solve your quadratic for that consant, will cosine x equal you're constant? The fact that you constant is a variable dependent on some function of x is what makes it not a quadratic, and likewise, makes your approach not work. I suggest you graph the functions x^2+x \mbox{and} cos x on graphs on top of eachother, and then try to find, using the information you'll learn through that, all (if any exist) of the real roots.
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  6. #5  
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    he fact that you constant is a variable dependent on some function of x is what makes it not a quadratic
    But, wait, you're not quite getting me. Perhaps I should clarify what precisely I have done in order to explain.

    I think you're thinking that I've allowed c itself to be a variable, which is not quite what I have done. I have only presented a generalized version of all the equations we'd have to solve to obtain the different values of Q; in the quadratic formula, I then proceed to substitute the end limits of c. C is, in fact, a constant; but the equations we have to solve for each constant value of c are different.

    To be precise, from the quadratic formula we know that



    (Apologies for that horrible mess somewhere in between)

    Obviously, b = a = 1 for every one of the equations that are generated. So,



    It should be clear that the values of Q depend directly on c. Thus, I should be permitted to place my end limits of c and solve separately.

    The equation I originally wrote is a generalised form of all the separate equations involving different values of c. The general value of Q is thus obtained, and solved for the two end limits. Real or malarkey?
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  7. #6  
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    Arcane's right here. You can't fix a single term and expect to get the right answer. For example, take c = 0. Q^2 + Q = 0 gives 0 or -1 as answers, but neither of those works for x^2 + x + cos x = 0.
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  8. #7  
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    Arcane's right here. You can't fix a single term and expect to get the right answer. For example, take c = 0. Q^2 + Q = 0 gives 0 or -1 as answers, but neither of those works for x^2 + x + cos x = 0.
    Very well, then. How do I solve this problem, then? I must admit, I'm stuck.
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  9. #8  
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    It might not be solvable. You can use Newton's method (or some other root finder) to get a numerical answer, but an exact symbolic answer might not exist.
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