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.